estimation for partially nonstationary multivariate autoregressive models abstract for an m variate partially nonstationary vector autoregressive process y t we consider the autoregressive model l y t e t where l i 1 l p l p and det z 0 has d m roots equal to unity and all other roots are outside the unit circle it is also assumed that rank 1 r r m d so that each component of the first differences w t y t y t 1 is stationary the relation of the model to error correction models and co integration engle and granger 1987 is discussed the process y t has the error correction representation l 1 l y t p 2 i r r q 2 1 y t 1 e t where q i m 1 p j diag i d r is in jordan canonical form and q q 1 q 2 it follows that the transformation z t qy t zt 1t z2 2t t is such that the d 1 process z 1t is nonstationary with z 1t z 1t 1 stationary while z 2t is stationary asymptotic distribution theory for least squares parameter estimators of the model is first considered a gaussia

forecasting multiple time series with one sided dynamic principal components abstractwe define one sided dynamic principal components odpc for time series as linear combinations of the present and past values of the series that minimize the reconstruction mean squared err

convolutional autoregressive models for functional time series abstract functional data analysis has became an increasingly popular class of problems in statistical research however functional data observed over time with serial dependence remains a less studied area motivated by bosq 2000 who first introduced the functional autoregressive models we propose a convolutional functional autoregressive model where the function at time t is a result of the sum of convolutions of the past functions and a set of convolution functions plus a noise process mimicking the vector autoregressive process it provides an intuitive and direct interpretation of the dynamics of a stochastic process instead of principal component analysis commonly used in functional data analysis we adopt a sieve estimation procedure based on b spline approximation of the convolution functions we establish convergence rate of the proposed estimator and investigate its theoretical properties the model building model validation and prediction procedures are also developed both simulated and real data examples are presented

modelling multiple time series via common factors we propose a new method for estimating common factors of multiple time series one distinctive feature of the new approach is that it is applicable to some nonstationary time series the unobservable nonstationary factors are identified by expanding the white noise space step by step thereby solving a high dimensional optimization problem by several low dimensional sub problems asymptotic properties of the estimation are investigated the proposed methodology is illustrated with both simulated and real datasets

the conditional autoregressive wishart model for multivariate stock market volatility abstract we propose a conditional autoregressive wishart caw model for the analysis of realized covariance matrices of asset returns our model assumes an autoregressive moving average structure for the scale matrix of the wishart distribution it accounts for positive definiteness of covariance matrices without imposing parametric restrictions and can be estimated by maximum likelihood we also propose extensions of the caw model obtained by including a mixed data sampling midas component and heterogeneous autoregressive har dynamics for long run fluctuations the caw models are applied to realized variances and covariances for five new york stock exchange stocks

riccati equations for abnormal time scale quadratic functionals abstract this paper focuses on developing new riccati type conditions for an abnormal time scale symplectic system s these conditions provide characterizations of the nonnegativity with and without a certain image condition and positivity of the quadratic functionals associated with such a system the novelty of these conditions rely on the natural conjoined basis x a u a of s in which x a t is not necessarily invertible and thus the system s could be abnormal these results are new even in the special case of continuous time as are some of them in the discrete time setting

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

an irregular sampling theorem for functions bandlimited in a generalized sense it is shown that a well known irregular sampling formula for ordinary bandlimited functions is also valid for functions bandlimited in a generalized sense except for some necessary modifications

use of canonical analysis in time series model identification summary the second order moment structure of time series models is used to derive a canonical analysis in time series modelling consistency properties of certain canonical correlations and the corresponding eigenvectors are shown based on these properties a canonical correlation approach for tentative order determination in building autoregressivemoving average models is proposed this approach can handle directly nonstationary as well as stationary processes and it also provides consistent estimates of the autoregressive parameters involved the asymptotic distribution of the identification statistic is discussed

modeling multiple time series with applications abstract an approach to the modeling and analysis of multiple time series is proposed properties of a class of vector autoregressive moving average models are discussed modeling procedures consisting of tentative specification estimation and diagnostic checking are outlined and illustrated by three real examples

uniformly lipschitz feedback optimal controls in a linear quadratic framework a uniformly k lipschitz feedback optimal control problem is considered in a linear quadratic framework the value function is derived by a comparison based reasoning via which a necessary condition to the existence of optimal solutions is obtained optimal feedback controls must be linear

rationality minimality and uniqueness of representation of matrix formal power series abstract in some multivariate time series models a matrix power series is involved these models can be identified as rational models if these series correspond to a matrix rational function moreover it is necessary to answer some questions about minimality and uniqueness of representation the main results of this paper fall within the sphere of matrix pade approximation on the basis of formal power series matrix rational functions of arbitrary dimensions are characterized furthermore we study certain minimality types that are global minimum degrees and row minimum degrees in addition given that the rational representation of the function for the same pair of degrees need not be unique we have obtained conditions to study the uniqueness of said representation and also to find a canonical unique representation moreover we consider an application to special series which is associated with time series models such series leads to new theoretical results relating to matrix pade approximation finally we comment on some illustrative examples

generating univariate fractional integration within a large var 1 abstract this paper shows that a large dimensional vector autoregressive model var of finite order can generate fractional integration in the marginalized univariate series we derive high level assumptions under which the final equation representation of a var 1 leads to univariate fractional white noises and verify the validity of these assumptions for two specific models

on likelihood ratio tests for threshold autoregression summary this paper addresses the null distribution of the likelihood ratio statistic for threshold autoregression with normally distributed noise the problem is non standard because the threshold parameter is a nuisance parameter which is absent under the null hypothesis we reduce the problem to the first passage probability associated with a gaussian process which in some special cases turns out to be a brownian bridge it is also shown that in some specific cases the asymptotic null distribution of the test statistic depends only on the degrees of freedom and not on the exact null joint distribution of the time series

coupled intervals in the discrete calculus of variations necessity and sufficiency in this work we study nonnegativity and positivity of a discrete quadratic functional with separately varying endpoints we introduce a notion of an interval coupled with 0 and hence extend the notion of conjugate interval to 0 from the case of fixed to variable endpoint s we show that the nonnegativity of the discrete quadratic functional is equivalent to each of the following conditions the nonexistence of intervals coupled with 0 the existence of a solution to riccati matrix equation and its boundary conditions natural strengthening of each of these conditions yields a characterization of the positivity of the discrete quadratic functional since the quadratic functional under consideration could be a second variation of a discrete calculus of variations problem with varying endpoints we apply our results to obtain necessary and sufficient optimality conditions for such problems this paper generalizes our recent work in r hilscher v zeidan comput math appl to appear where the right endpoint is fixed

studying co movements in large multivariate data prior to multivariate modelling abstract for non stationary vector autoregressive models var hereafter or var with moving average varma hereafter we show that the presence of common cyclical features or cointegration leads to a reduction of the order of the implied univariate autoregressive integrated moving average arima hereafter models this finding can explain why we identify parsimonious univariate arima models in applied research although var models of typical order and dimension used in macroeconometrics imply non parsimonious univariate arima representations next we develop a strategy for studying interactions between variables prior to possibly modelling them in a multivariate setting indeed the similarity of the autoregressive roots will be informative about the presence of co movements in a set of multiple time series our results justify both the use of a panel setup with homogeneous autoregression and heterogeneous cross correlated vector moving average errors and a factor structure and the use of cross sectional aggregates of arima series to estimate the homogeneous autoregression

a characterization of vector autoregressive processes with common cyclical features this paper presents necessary and sufficient conditions for the existence of common cyclical features in vector auto regressive var processes integrated of order 0 1 2 where the common cyclical features correspond to common serial correlation cs commonality in the final equations ce and co dependence cd the results are based on local rank factorizations of the reversed ar polynomial around the poles of its inverse all processes with cs structures are found to present also ce structures and vice versa the presence of cd structures instead implies the presence of both cs and ce structures but not vice versa characterizations of the cs ce cd linear combinations are given in terms of linear subspaces defined in the local rank factorizations

martingale difference divergence matrix and its application to dimension reduction for stationary multivariate time series in this article we introduce a new methodology to perform dimension reduction for a stationary multivariate time series our method is motivated by the consideration of optimal prediction and focu

a tabular methodology to identify minimal row degrees for matrix pade approximants in this paper we propose a tabular procedure for identifying sets of minimal row degrees corresponding to matrix pade approximants the objective is to make it easier to interpret and properly apply them in various fields when considering formal matrix power series with kxm coefficients it is useful to examine the linearly dependent rows that are in the last k rows in certain hankel matrices corresponding to the coefficients of the series a list of properties and suitable examples are added not only by way of illustration but also because they are an original part of the method having been chosen to teach the procedure and to underscore certain precautions to be taken

some results on multivariate autoregressive index models summary we discuss methods for modelling multivariate autoregressive time series in terms of a smaller number of index series which are chosen to provide as complete a summary as possible of the past information contained in the original series necessary for prediction purposes the maximum likelihood method of estimation and asymptotic properties of estimators of the coefficients which determine the index variables as well as the corresponding autoregressive coefficients are discussed a numerical example is presented to illustrate the use of the autoregressive index models

dynamic orthogonal components for multivariate time series we introduce dynamic orthogonal components doc for multivariate time series and propose a procedure for estimating and testing the existence of docs for a given time series we estimate the dynamic orthogonal components via a generalized decorrelation method that minimizes the linear and quadratic dependence across components and across time we then use ljung box type statistics to test the existence of dynamic orthogonal components when docs exist univariate analysis can be applied to build a model for each component those univariate models are then combined to obtain a multivariate model for the original time series we demonstrate the usefulness of dynamic orthogonal components with two real examples and compare the proposed modeling method with other dimension reduction methods available in the literature including principal component and independent component analyses we also prove consistency and asymptotic normality of the proposed estimator under some regularity conditions we provide some

identifying a simplifying structure in time series abstract this article studies how to identify hidden factors in multivariate time series process this problem is important because when the series are driven by a set of common factors a a large number of parameters may be needed to obtain an adequate representation of the system and b the estimated parameters will be highly correlated therefore a complex and badly defined relationship can appear when in fact a simpler and parsimonious model in terms of a few common factors can be operating this article develops a methodology to identify the number of factors and to build a simplifying transformation to represent the series it is proved that the number of factors is equal to the rank of the covariance matrices and the parameter matrices of the infinite moving average representation of the process the eigenvectors of these matrices will provide the canonical transformation the method is illustrated with one example using series of the price of wheat in five provinces of spain in the 19th ce

high dimensional stochastic regression with latent factors endogeneity and nonlinearity abstract we consider a multivariate time series model which represents a high dimensional vector process as a sum of three terms a linear regression of some observed regressors a linear combination of some latent and serially correlated factors and a vector white noise we investigate the inference without imposing stationary conditions on the target multivariate time series the regressors and the underlying factors furthermore we deal with the endogeneity that there exist correlations between the observed regressors and the unobserved factors we also consider the model with nonlinear regression term which can be approximated by a linear regression function with a large number of regressors the convergence rates for the estimators of regression coefficients the number of factors factor loading space and factors are established under the settings when the dimension of time series and the number of regressors may both tend to infinity together with the sample size the proposed method is illustrated with both simulated and real data examples

encompassing univariate models in multivariate time series a case study abstract through the encompassing principle univariate arima analysis could provide an important tool for diagnosis of var models the univariate arima models implied by the var should explain the results from univariate analysis this comparison is seldom performed possibly due to the paradox that while the implied arima models typically contain a very large number of parameters univariate analysis yields highly parsimonious models using a var application to six french macro economic variables it is seen how the encompassing check is straight forward to perform and surprisingly accurate

dynamic factor analysis of nonstationary multivariate time series a dynamic factor model is proposed for the analysis of multivariate nonstationary time series in the time domain the nonstationarity in the series is represented by a linear time dependent mean function this mild form of nonstationarity is often relevant in analyzing socio economic time series met in practice through the use of an extended version of molenaar s stationary dynamic factor analysis method the effect of nonstationarity on the latent factor series is incorporated in the dynamic nonstationary factor model dnfm it is shown that the estimation of the unknown parameters in this model can be easily carried out by reformulating the dnfm as a covariance structure model and adopting the ml algorithm proposed by joreskog furthermore an empirical example is given to demonstrate the usefulness of the proposed dnfm and the analysis

forecasting with nonstationary dynamic factor models abstract in this paper we analyze the structure and the forecasting performance of the dynamic factor model it is shown that the forecasts obtained by the factor model imply shrinkage pooling terms similar to the ones obtained from hierarchical bayesian models that have been applied successfully in the econometric literature thus the results obtained in this paper provide an additional justification for these and other types of pooling procedures the expected decrease in mse for using a factor model versus univariate arima and shrinkage models are studied for the one factor model monte carlo simulations are presented to illustrate this result a factor model is also built to forecast gnp of european countries and it is shown that the factor model can provide a substantial improvement in forecasts with respect to both univariate and shrinkage univariate forecasts

large sample estimation and testing procedures for dynamic equation systems abstract joint two step estimation procedures which have the same asymptotic properties as the maximum likelihood ml estimator are developed for the final equation transfer function and structural form of a multivariate dynamic model with normally distributed vector moving average errors the ml estimator under fixed and known initial values is obtained by iterating the procedure until convergence the asymptotic distribution of the two step estimators is used to construct large sample testing procedures for the different forms of the model

dimension reduction in time series and the dynamic factor model this note shows that the dimension reduction method proposed by li shedden 2002 is equivalent to the dynamic factor model introduced by pena box 1987

the regulator problem with indefinite quadratic cost for boundary control systems the finite horizon case in this paper we study the finite horizon non standard lq problem for an abstract dynamics which models a large class of hyperbolic like partial differential equations we provide necessary sufficient conditions for finiteness of the value function corresponding to the control problem sharpness of sufficient conditions is shown by means of counterexamples the specific features of the finite in contrast to infinite horizon case are illustrated

sturm liouville problems with parameter dependent potential and boundary conditions this paper deals with the computation of the eigenvalues of sturm liouville problems with parameter dependent potential and boundary conditions we shall extend the domain of application of the method based on sampling theory to the case where the classical whittaker shannon kotel nikov theorem is not applicable a few numerical examples will be presented

a type of matrix pade approximant inspired by scalar component models in this paper we define a type of matrix pade approximant inspired by the identification stage of multivariate time series models considering scalar component models of course the formalization of certain properties in the matrix pade approximation framework can be applied to time series models and in other fields specifically we want to study matrix pade approximants as follows to find rational representations or rational approximations of a matrix formal power series with both matrix polynomials numerator and denominator satisfying three conditions a minimum row degrees for the numerator and denominator b an invertible denominator at the origin and c canonical representation without free parameters

general sampling theorem using contour integral abstract we present the sampling theorem with sampling functions of general form for entire functions satisfying one of the growth conditions 1 y f z a 1 z n 1 exp x y 1 x f z a 1 z n 1 exp x y for some a 0 0 n 1 n 0 and any z x iy c it will be shown that many well known sampling theorems included in siam j math anal 19 1988 1198 1203 and inform control 8 1965 143 158 can be interpreted as special cases of this sampling theorem as examples we provide sampling representations for entire functions which are bounded of polynomial growth or of exponential growth on r we also provide sampling representations involving derivatives of entire functions and nonuniform sampling representations taking the set of sampling points in which a finite number of points are arbitrarily distributed we obtain a sampling representation

reduced rank models for multiple time series summary this paper is concerned with the investigation of reduced rank coefficient models for multiple time series in particular autoregressive processes which have a structure to their coefficient matrices similar to that of classical multivariate reduced rank regression are studied in detail the estimation of parameters and associated asymptotic theory are derived the exact correspondence between the reduced rank regression procedure for multiple autoregressive processes and the canonical analysis of box tiao 1977 is briefly indicated to illustrate the methods u s hog data are considered

outliers in multivariate time series summary this paper generalises four types of disturbance commonly used in univariate time series analysis to the multivariate case highlights the differences between univariate and multivariate outliers and investigates dynamic effects of a multivariate outlier on individual components the effect of a multivariate outlier depends not only on its size and the underlying model but also on the interaction between the size and the dynamic structure of the model the latter factor does not appear in the univariate case a multivariate outlier can introduce various types of outlier for the marginal component models by comparing and contrasting results of univariate and multivariate outlier detections one can gain insights into the characteristics of an outlier we use real examples to demonstrate the proposed analysis

generalized dynamic principal components abstractbrillinger defined dynamic principal components dpc for time series based on a reconstruction criterion he gave a very elegant theoretical solution and proposed an estimator which is consistent under stationarity here we propose a new enterally empirical approach to dpc the main differences with the existing methods mainly brillinger procedure are 1 the dpc we propose need not be a linear combination of the observations and 2 it can be based on a variety of loss functions including robust ones unlike brillinger we do not establish any consistency results however contrary to brillinger s which has a very strong stationarity flavor our concept aims at a better adaptation to possible nonstationary features of the series we also present a robust version of our procedure that allows to estimate the dpc when the series have outlier contamination we give iterative algorithms to compute the proposed procedures that can be used with a large number of variables our nonrobust and robust pr

a bayesian multivariate nonstationary time series model for estimating mutual relationships among variables abstract the purpose of this paper is to propose a bayesian multivariate stochastic model with latent nonstationary trends and seasonal components and show its use to determine the relationships among the variables the model is expressed in state space form and the parameters of the model are estimated by maximum likelihood using a numerical optimization algorithm the kalman filter is used to compute the likelihood of the model and the information criterion aic is used to select the best fitting model the relationships among variables are examined in the frequency domain using estimated components japanese macroeconomic series are analyzed by our procedure

accurate solutions of fourth order sturm liouville problems recently we introduced a new method which we call the extended sampling method to compute the eigenvalues of second order sturm liouville problems with eigenvalue dependent potential we shall see in this paper how we use this method to compute the eigenvalues of fourth order sturm liouville problems and present its practical use on a few examples

a unified approach to identifying multivariate time series models abstract this article proposes a bayesian procedure for simultaneous identification of the kronecker indices and model parameters of a multivariate linear system the model parameters include the starting values and innovations of the system so that the series considered may be co integrated or non invertible the procedure uses some recent developments in stochastic search variable selection in linear regression analysis and markov chain monte carlo methods in statistical computing it also takes into consideration the row structure of a vector model implied by the kronecker indices comparison with other existing methods is discussed simulated and real examples are used to illustrate the proposed procedure

nested reduced rank autoregressive models for multiple time series abstract the nested reduced rank autoregressive ar model is considered in order to simplify and provide a more detailed description of the structure of the multivariate time series and to reduce the number of parameters in the time series modeling the multivariate ar model is yt p j 1 j y t j et where yt is m 1 and the structure of the model considered is such that the rj rank j are nonincreasing as the lag j increases so the j have the factorization j ajbj and range aj range a j 1 specification of the coefficient rank structures for such models through the use of canonical correlation analysis between yk t y t y t k and y k t 1 is discussed a canonical variable transformation that produces simpler structure in the model and explicitly illustrates how different components of the vector series depend on past lagged values to differing degrees is also examined a gaussian parameter estimation procedure is described and asymptotic properties of the gaussian estim

large bayesian varmas vector autoregressive moving average varma models have many theoretical properties which should make them popular among empirical macroeconomists however they are rarely used in practice due to over parameterization concerns difficulties in ensuring identification and computational challenges with the growing interest in multivariate time series models of high dimension these problems with varmas become even more acute accounting for the dominance of vars in this field in this paper we develop a bayesian approach for inference in varmas which surmounts these problems it jointly ensures identification and parsimony in the context of an efficient markov chain monte carlo mcmc algorithm we use this approach in a macroeconomic application involving up to twelve dependent variables we find our algorithm to work successfully and provide insights beyond those provided by vars

asymptotic properties of multivariate nonstationary processes with applications to autoregressions asymptotic properties of multivariate time series with characteristic roots on the unit circle are considered for a vector autoregressive moving average process we derive the limiting distributions of certain statistics which are useful in understanding nonstationary processes these distributions are derived in a unified manner for all types of characteristic roots and are expressed in terms of stochastic integrals of brownian motions we use the limiting distributions to establish the consistency properties of the ordinary least squares estimates of various autoregressions of a vector process

constrained regular lq control problems in this paper we give a complete description of how the jacobi theory of conjugate points can be extended to a regular linear quadratic control problem where the state end points are jointly constrained to belong to a subspace of bf r n times bf r n and there is a linear pointwise state control constraint we introduce also the definition of semiconjugate point which describes a distinctive feature of these problems and state the corresponding necessary and sufficient conditions for the quadratic form to be nonnegative or coercive in the case in which the constraints and the costs act separately on the initial and final points we give equivalent characterizations of the coercivity of the quadratic form by means of the solutions of an associated riccati equation in both the controllable and uncontrollable case

oscillation theorems and rayleigh principle for linear hamiltonian and symplectic systems with general boundary conditions the aim of this paper is to establish the oscillation theorems rayleigh principle and coercivity results for linear hamiltonian and symplectic systems with general boundary conditions i e for the case of separated and jointly varying endpoints and with no controllability normality and strong observability assumptions our method is to consider the time interval as a time scale and apply suitable time scales techniques to reduce the problem with separated endpoints into a problem with dirichlet boundary conditions and the problem with jointly varying endpoints into a problem with separated endpoints these more general results on time scales then provide new results for the continuous time linear hamiltonian systems as well as for the discrete symplectic systems this paper also solves an open problem of deriving the oscillation theorem for problems with periodic boundary conditions furthermore the present work demonstrates the utility and power of the analysis on time scales in obtaining new results especially in the classical continuous and discrete time theories

the importance of common cyclical features in var analysis a monte carlo study abstract despite the commonly held belief that aggregate data display short run comovement there has been little discussion about the econometric consequences of this feature of the data we use exhaustive monte carlo simulations to investigate the importance of restrictions implied by common cyclical features for estimates and forecasts based on vector autoregressive models first we show that the best empirical model developed without common cycle restrictions need not nest the best model developed with those restrictions this is due to possible differences in the lag lengths chosen by model selection criteria for the two alternative models second we show that the costs of ignoring common cyclical features in vector autoregressive modelling can be high both in terms of forecast accuracy and efficient estimation of variance decomposition coefficients third we find that the hannan quinn criterion performs best among model selection criteria in simultaneously selecting the lag length and rank of vector autoregressions