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

partial non gaussian state space summary in this paper we suggest the use of simulation techniques to extend the applicability of the usual gaussian state space filtering and smoothing techniques to a class of nongaussian time series models this allows a fully bayesian or maximum likelihood analysis of some interesting models including outlier models discrete markov chain components multiplicative models and stochastic variance models finally we discuss at some length the use of a non gaussian model to seasonally adjust the published money supply figures

fourier integral operators fractal sets and the regular value theorem we prove that if e r2d for d 2 is an ahlfors david regular product set of sufficiently large hausdorff dimension denoted by dimh e and is a sufficiently regular function then the upper minkowski dimension of the set w e l w tl 1 l m does not exceed dimh e m in line with the regular value theorem from the elementary differential geometry our arguments are based on the mapping properties of the underlying fourier integral operators and are intimately connected with the falconer distance conjecture in geometric measure theory we shall see that our results are in general sharp in the sense that if the hausdorff dimension is smaller than a certain threshold then the dimensional inequality fails in a quantifiable way the constructions used to demonstrate this are based on the distribution of lattice points on convex surfaces and have connections with combinatorial geometry

a converse to the deterministic separation principle in the classical theory of finite dimensional linear time invariant systems in state space form the term deterministic separation principle refers to the observation that a stabilizing output feedback controller can be constructed by first constructing an asymptotic state observer that is then coupled to a stabilizing state feedback controller in this paper we discuss the following converse problem can every stabilizing output feedback controller be realized as interconnection of an asymptotic state observer and a stabilizing state feedback controller we will provide an affirmative answer to this question modulo a number of technicalities in a behavioral setting and with the help of rational representations

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

distance between behaviors and rational representations in this paper we study notions of distance between behaviors of linear differential systems we introduce four metrics on the space of all controllable behaviors which generalize existing metrics on the space of input output systems represented by transfer matrices three of these are defined in terms of gaps between closed subspaces of the hilbert space mathcal l 2 mathbb r in particular we generalize the classical gap metric we express these metrics in terms of rational representations of behaviors in order to do so we establish a precise relation between rational representations of behaviors and multiplication operators on mathcal l 2 mathbb r we introduce a fourth behavioral metric as a generalization of the well known nu metric as in the input output framework this definition is given in terms of rational representations for this metric however we establish a representation free behavioral characterization as well we make a comparison between the four metrics and compar

input output and state space representations of finite dimensional linear time invariant systems in this paper we develop various representations for systems described by aset of high order differential equations of the form r0w r1w rsw s 0 with r0 r1 rs not necessarily square matrices the variables w are the external variables particular attention is paid to the problem of obtaining minimal state space realizations and input output or input state output representations of such systems

temporal aggregation of garch processes the authors derive low frequency say weekly models implied by high frequency say daily arma models with symmetric garch errors they show that low frequency models exhibit conditional heteroskedasticity of the garch form as well the parameters in the conditional variance equation of the low frequency model depend upon mean variance and kurtosis parameters of the corresponding high frequency model moreover strongly consistent estimators of the parameters in the high frequency model can be derived from low frequency data the common assumption in applications that rescaled innovations are independent is disputable since it depends upon the available data frequency copyright 1993 by the econometric society

transformations of linear systems under external equivalence abstract we consider systems of linear differential and algebraic equations in which some of the variables are distinguished as external variables two systems are called equivalent if the set of solutions for the external variables is the same for both systems we give an operational form for this definition of equivalence i e we describe a set of system transformations having the property that two systems are equivalent if and only if they can be taken into each other by transformations from that set next an algorithm is described to transform a given system in general form to a system in minimal state space form this algorithm differs from existing methods in that it first takes the equations to first order form so that each subsequent step can be formulated and interpreted in state space terms we also compute the structure indices in terms of a state space description in nonminimal form and use this to prove the minimality of the end result of the algorithm finally an application is shown to the problem of ill posedness of feedback connections

on gibbs sampling for state space models summary we show how to use the gibbs sampler to carry out bayesian inference on a linear state space model with errors that are a mixture of normals and coefficients that can switch over time our approach simultaneously generates the whole of the state vector given the mixture and coefficient indicator variables and simultaneously generates all the indicator variables conditional on the state vectors the states are generated efficiently using the kalman filter we illustrate our approach by several examples and empirically compare its performance to another gibbs sampler where the states are generated one at a time the empirical results suggest that our approach is both practical to implement and dominates the gibbs sampler that generates the states one at a time

likelihood based estimation of latent generalized arch structures garch models are commonly used as latent processes in econometrics financial economics and macroeconomics yet no exact likelihood analysis of these models has been provided so far in this paper we outline the issues and suggest a markov chain monte carlo algorithm which allows the calculation of a classical estimator via the simulated em algorithm or a bayesian solution in o t computational operations where t denotes the sample size we assess the performance of our proposed algorithm in the context of both artificial examples and an empirical application to 26 uk sectorial stock returns and compare it to existing approximate solutions

marginalization and contemporaneous aggregation in multivariate garch processes abstract we first of all show that contemporaneous aggregation of independent univariate garch processes yields a weak garch process as introduced by drost and nijman 1993 subsequently we analyze the dependence of the parameters in the aggregate on the parameters in the underlying models and show that the variance parameters after aggregation depend on the underlying variance and kurtosis parameters then we generalize the results by showing that a linear combination of variables generated by a multivariate garch process will also be weak garch finally we derive the marginal weak garch processes implied by several multivariate garch processes

factor representing portfolios in large asset markets we study the properties of mimicking portfolios in an intertemporal apt model in which the conditional mean and covariance matrix of returns vary in an interdependent manner we use a signal extraction approach and relate the efficiency of possibly dynamic basis portfolios to mean square error minimisation we prove that many portfolios converge to the factors as the number of assets increases but show that the conditional kalman filter portfolios are the ones with both minimum tracking error variability and maximum correlation with the common factors we also show that our conclusions are unlikely to change when using parameter estimates

on quadratic differential forms this paper develops a theory around the notion of quadratic differential forms in the context of linear differential systems in many applications we need to not only understand the behavior of the system variables but also the behavior of certain functionals of these variables the obvious cases where such functionals are important are in lyapunov theory and in lq and h infty optimal control with some exceptions these theories have almost invariably concentrated on first order models and state representations in this paper we develop a theory for linear time invariant differential systems and quadratic functionals we argue that in the context of systems described by one variable polynomial matrices the appropriate tool to express quadratic functionals of the system variables are two variable polynomial matrices the main achievement of this paper is a description of the interaction of one and two variable polynomial matrices for the analysis of functionals and for the application of higher order lyapunov functionals

on the gap metric and coprime factor perturbations abstract new conditions are derived for when the distance between two linear systems in the gap metric is less than one by including a coprimeness assumption in the coprime factor uncertainty description it is shown that an open ball in the gap metric is equivalent to an open ball phrased in terms of coprime factor perturbations a new criterion is given for robust stabilization

fractional dimension of sets in discrete spaces the authors give a new definition dimh a for the dimension of an arbitrary subset of the lattice d they establish elementary properties and calculate the dimension for some examples finally they announce a result which states that if dimh a d 2 then a is recurrent

behaviors defined by rational functions abstract in this article behaviors defined by differential equations involving matrices of rational functions are introduced conditions in terms of controllability observability and stabilizability for the existence of rational representations that are prime over various subrings of the field of rational functions are derived elimination of latent variables observable image like representations of controllable systems and the structure of the rational annihilators of a behavior are discussed

distance and tube zeta functions of fractals and arbitrary compact sets abstract recently the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets a of the n dimensional euclidean space r n for any integer n 1 it is defined by the lebesgue integral a s a d x a s n d x for all s c with re s sufficiently large and we call it the distance zeta function of a here d x a denotes the euclidean distance from x to a and a is the neighborhood of a where is a fixed positive real number we prove that the abscissa of absolute convergence of a is equal to dim b a the upper box or minkowski dimension of a particular attention is payed to the principal complex dimensions of a defined as the set of poles of a located on the critical line re s dim b a provided a possesses a meromorphic extension to a neighborhood of the critical line we also introduce a new closely related zeta function a s 0 t s n 1 a t d t called the tube zeta function of a assuming that a is minkowski measurable we show that under some mild conditions the residue of a computed at d dim b a the box dimension of a is equal to the minkowski content of a more generally without assuming that a is minkowski measurable we show that the residue is squeezed between the lower and upper minkowski contents of a we also introduce transcendentally quasiperiodic sets and construct a class of such sets using generalized cantor sets along with baker s theorem from the theory of transcendental numbers

factor garch ito models for high frequency data with application to large volatility matrix prediction abstract several novel large volatility matrix estimation methods have been developed based on the high frequency financial data they often employ the approximate factor model that leads to a low rank plus sparse structure for the integrated volatility matrix and facilitates estimation of large volatility matrices however for predicting future volatility matrices these nonparametric estimators do not have a dynamic structure to implement in this paper we introduce a novel ito diffusion process based on the approximate factor models and call it a factor garch ito model we then investigate its properties and propose a quasi maximum likelihood estimation method for the parameter of the factor garch ito model we also apply it to estimating conditional expected large volatility matrices and establish their asymptotic properties simulation studies are conducted to validate the finite sample performance of the proposed estimation methods the proposed method is also illustrated by using data from the constituents of the s p 500 index and an application to constructing the minimum variance portfolio with gross exposure constraints

stability of control systems and graphs of linear systems new conditions for internal stability of a closed loop control system are given in terms of the graphs of the multiplication operators induced by the transfer functions of the plant and the controller these conditions can be given a geometrical interpretation this relates closed loop stability to the minimal angle between the graph space associated with the system and the graph space associated with the controller the maximally stabilizing controller is defined as the controller that maximizes the minimum angle between the graph space associated with the system and the graph space associated with the controller it is shown that this controller can be calculated as a nehari extension of the coprime factors of the system

equivalence of rational representations of behaviors this article deals with the equivalence of representations of behaviors of linear differential systems in general the behavior of a given linear differential system has many different representations in this paper we restrict ourselves to kernel and image representations two kernel representations are called equivalent if they represent one and the same behavior for kernel representations defined by polynomial matrices necessary and sufficient conditions for equivalence are well known in this paper we deal with the equivalence of rational representations i e kernel and image representations that are defined in terms of rational matrices as the first main result of this paper we will derive a new condition for the equivalence of rational kernel representations of possibly noncontrollable behaviors secondly we will derive conditions for the equivalence of rational representations of a given behavior in terms of the polynomial modules generated by the rows of the rational matrices we will also establish conditions for the equivalence of rational image representations finally we will derive conditions under which a given rational kernel representation is equivalent to a given rational image representation

zeta functions of discrete self similar sets abstract in this paper we study a class of countable and discrete subsets of a euclidean space that are self similar with respect to a finite set of affine similarities any such set can be interpreted as having a fractal structure we introduce a zeta function for these sets and derive basic analytic properties of this fractal zeta function motivating examples that come from combinatorial geometry and arithmetic are given particular attention

complex dimensions of fractals and meromorphic extensions of fractal zeta functions abstract we study meromorphic extensions of distance and tube zeta functions as well as of geometric zeta functions of fractal strings the distance zeta function a s a d x a s n d x where 0 is fixed and d x a denotes the euclidean distance from x to a has been introduced by the first author in 2009 extending the definition of the zeta function l associated with bounded fractal strings l l j j 1 to arbitrary bounded subsets a of the n dimensional euclidean space the abscissa of lebesgue i e absolute convergence d a coincides with d dim b a the upper box or minkowski dimension of a the visible complex dimensions of a are the poles of the meromorphic continuation of the fractal zeta function i e the distance or tube zeta function of a to a suitable connected neighborhood of the critical line re s d we establish several meromorphic extension results assuming some suitable information about the second term of the asymptotic expansion of the tube function a t as t 0 where a t is the euclidean t neighborhood of a we pay particular attention to a class of minkowski measurable sets such that a t t n d m o t as t 0 with 0 and to a class of minkowski nonmeasurable sets such that a t t n d g log t 1 o t as t 0 where g is a nonconstant periodic function and 0 in both cases we show that a can be meromorphically extended at least to the open right half plane re s d and determine the corresponding visible complex dimensions furthermore up to a multiplicative constant the residue of a evaluated at s d is shown to be equal to m the minkowski content of a and to the mean value of g the average minkowski content of a respectively moreover we construct a class of fractal strings with principal complex dimensions of any prescribed order as well as with an infinite number of essential singularities on the critical line re s d finally using an appropriate quasiperiodic version of the above construction with infinitely many suitably chosen quasiperiods associated with a two parameter family of generalized cantor sets we construct maximally hyperfractal compact subsets of r n for n 1 arbitrary these are compact subsets of r n such that the corresponding fractal zeta functions have nonremovable singularities at every point of the critical line re s d

zeta functions and solutions of falconer type problems for self similar subsets of ℤn this paper uses the method of zeta functions to solve falconer type problems about sets of k simplices whose endpoints belong simultaneously to a self similar subset f of z k n and a disc b x of a large radius x assuming that the similarity transformations pairwise commute we study four euclidean metric invariants of the simplices the most basic and frequently studied of which is the distance between endpoints of a 1 simplex for each we introduce a zeta function derive its functional properties and apply such information to derive a lower bound on the upper minkowski dimension of f which guarantees that the number of distinct metric invariants must be unbounded as x mathematics subject classifications 11m41 28a80 52c10 11j69 11m32

realization and elimination in rational representations of behaviors abstract this article deals with the relationship between rational representations of linear differential systems and their state representations in particular we study the relationship between rational representations on the one hand and output nulling and driving variable representations on the other in the input output framework it is well known that every controllable and observable realization of the transfer matrix of the system yields a minimal input state output representation if a proper rational matrix is used for a rational kernel or image representation of the system then the question arises under what conditions realizations of this rational matrix give rise to state representations we will establish conditions under which realizations of the proper rational matrix appearing in a rational image representation yield minimal driving variable representations of the system likewise we find conditions under which realization leads from rational kernel representations to minimal output nulling representations we also study the converse problem namely state elimination from driving variable and output nulling representations we will establish closed form expressions for kernel and image representations of the external behaviors associated with these particular state representations

unobserved component time series models with arch disturbances abstract this paper considers how arch effects may be handled in time series models formulated in terms of unobserved components a general model is formulated but this includes as special cases a random walk plus noise model with both disturbances subject to arch effects an arch m model with a time varying parameter and a latent factor model with arch effects in the factors although the model is not conditionally gaussian an approximate filter can be obtained and used as the basis for estimation the performance of this method is examined on real data sets and monte carlo experiments are carried out the method is extended to handle garch and a latent factor model based on the t distribution

em algorithms for ml factor analysis the details of em algorithms for maximum likelihood factor analysis are presented for both the exploratory and confirmatory models the algorithm is essentially the same for both cases and involves only simple least squares regression operations the largest matrix inversion required is for aq q symmetric matrix whereq is the matrix of factors the example that is used demonstrates that the likelihood for the factor analysis model may have multiple modes that are not simply rotations of each other such behavior should concern users of maximum likelihood factor analysis and certainly should cast doubt on the general utility of second derivatives of the log likelihood as measures of precision of estimation

rational differential operators and their kernels abstract a linear differential integral operator is associated to a rational matrix in a natural way and the behavior is defined to be the kernel of this operator conditions under which two rational matrices have the same behavior are derived

analysis of high dimensional multivariate stochastic volatility models this paper is concerned with the bayesian estimation and comparison of flexible high dimensional multivariate time series models with time varying correlations the model proposed and considered here combines features of the classical factor model with that of the heavy tailed univariate stochastic volatility model a unified analysis of the model and its special cases is developed that encompasses estimation filtering and model choice the centerpieces of the estimation algorithm which relies on mcmc methods are 1 a reduced blocking scheme for sampling the free elements of the loading matrix and the factors and 2 a special method for sampling the parameters of the univariate sv process the resulting algorithm is scalable in terms of series and factors and simulation efficient methods for estimating the log likelihood function and the filtered values of the time varying volatilities and correlations are also provided the performance and effectiveness of the inferential methods are extensively tested using simulated data where models up to 50 dimensions and 688 parameters are fit and studied the performance of our model in relation to various multivariate garch models is also evaluated using a real data set of weekly returns on a set of 10 international stock indices we consider the performance along two dimensions the ability to correctly estimate the conditional covariance matrix of future returns and the unconditional and conditional coverage of the 5 and 1 value at risk var measures of four pre defined portfolios

identification estimation and testing of conditionally heteroskedastic factor models we investigate the effects of dynamic heteroskedasticity on statistical factor analysis we show that identification problems are alleviated when variation in factor variances is accounted for our results apply to dynamic apt models and other structural models we also find that traditional ml estimation of unconditional variance parameters remains consistent if the factor loadings are identified from the unconditional distribution but their standard errors must be robustified we develop a simple preliminary lm test for arch effects in the common factors and discuss two step consistent estimation of the conditional variance parameters finally we conduct a detailed simulation exercise

rational representations and controller synthesis of l2 behaviors this paper considers linear dynamical systems restricted to square integrable trajectories following the behavioral formalism a number of relevant classes of linear and shift invariant l 2 systems are defined it is shown that rational functions analytic in specific half spaces of the complex plane prove most useful for representing such systems for various classes of l 2 systems this paper provides a complete characterization of system equivalence in terms of rational kernel representations of l 2 systems in addition a complete solution is given for the problem when selected non manifest variables of an l 2 system can be completely eliminated from their behavior this elimination theorem has considerable independent interest in general modeling problems it is shown that the elimination result is key in the solution of the problem for realizing an arbitrary l 2 system as the interconnection of a given l 2 system and a to be synthesized l 2 system in the context of control this problem amounts to characterizing the existence and parameterization of all controllers that after interconnection with a given plant constitute a desired controlled system

behaviors and symbols of rational matrices abstract recently we have defined rational differential equations and derived conditions when two such equations have the same solution set in this article we study the same question but present a different approach based on the new notion of symbols also we try to describe the relationship with the work behaviors defined by rational functions of willems and yamamoto

temporal aggregation of volatility models in this paper we consider temporal aggregation of volatility models we introduce semiparametric volatility models termed square root stochastic autoregressive volatility sr sarv which are characterized by autoregressive dynamics of the stochastic variance our class encompasses the usual garch models and various asymmetric garch models moreover our stochastic volatility models are characterized by multiperiod conditional moment restrictions in terms of observables the sr sarv class is a natural extension of the class of weak garch models this extension has four advantages i we do not assume that fourth moments are finite ii we allow for asymmetries skewness leverage effect that are excluded from weak garch models iii we derive conditional moment restrictions and iv our framework allows us to study temporal aggregation of igarch models

internal model principles for observers this paper deals with the observer problem for dynamical systems in a behavioral context we are given a dynamical system together with a partition of the system variables into a set of known or measured variables and a set of unknown to be estimated variables the observer problem is to find a system that produces an estimate of the unknown variables on the basis of the known or measured variables for a given plant and partition we establish a characterization of all error behaviors that can be achieved by interconnecting the plant with some observer the main result of this paper is a very general behavioral formulation of an internal model principle for observers we will show that a nonintrusive observer achieves a stable error behavior if and only if in addition to a detectability condition on the observer the observer behavior contains the anti stabilizable part of the plant behavior

indirect estimation of large conditionally heteroskedastic factor models with an application to the dow 30 stocks we derive indirect estimators of conditionally heteroskedastic factor models in which the volatilities of common and idiosyncratic factors depend on their past unobserved values by calibrating the score of a kalman filter approximation with inequality constraints on the auxiliary model parameters we also propose alternative indirect estimators for large scale models and explain how to apply our procedures to many other dynamic latent variable models we analyse the small sample behaviour of our indirect estimators and several likelihood based procedures through an extensive monte carlo experiment with empirically realistic designs finally we apply our procedures to weekly returns on the dow 30 stocks