variation and efficiency of high frequency betas abstract this paper studies the efficient estimation of betas from high frequency return data on a fixed time interval under an assumption of equal diffusive and jump betas we derive the semiparametric efficiency bound for estimating the common beta and develop an adaptive estimator that attains the efficiency bound we further propose a hausman type test for deciding whether the common beta assumption is true from the high frequency data in our empirical analysis we provide examples of stocks and time periods for which a common market beta assumption appears true and ones for which this is not the case we further quantify empirically the gains from the efficient common beta estimation developed in the paper

on the equations and classification of toric quiver varieties toric quiver varieties moduli spaces of quiver representations are studied given a quiver and a weight there is an associated quasi projective toric variety together with a canonical embedding into projective space it is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most 3 in each fixed dimension d up to isomorphism there are only finitely many d dimensional toric quiver varieties a procedure for their classification is outlined

numerical computation of minimal polynomial bases a generalized resultant approach we propose a new algorithm for the computation of a minimal polynomial basis of the left kernel of a given polynomial matrix f s the proposed method exploits the structure of the left null space of generalized wolovich or sylvester resultants to compute row polynomial vectors that form a minimal polynomial basis of left kernel of the given polynomial matrix the entire procedure can be implemented using only orthogonal transformations of constant matrices and results to a minimal basis with orthonormal coefficients

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

estimation of integrated quadratic covariation with endogenous sampling times abstract when estimating high frequency covariance quadratic covariation of two arbitrary assets observed asynchronously simple assumptions such as independence are usually imposed on the relationship between the prices process and the observation times in this paper we introduce a general endogenous two dimensional nonparametric model because an observation is generated whenever an auxiliary process called observation time process hits one of the two boundary processes it is called the hitting boundary process with time process hbt model we establish a central limit theorem for the hayashi yoshida hy estimator under hbt in the case where the price process and the observation price process follow a continuous ito process we obtain an asymptotic bias we provide an estimator of the latter as well as a bias corrected hy estimator of the high frequency covariance in addition we give a consistent estimator of the associated standard error

fisher s information for discretely sampled levy processes this paper studies the asymptotic behavior of the fisher information for a levy process discretely sampled at an increasing frequency we show that it is possible to distinguish not only the continuous part of the process from its jumps part but also different types of jumps and derive the rates of convergence of efficient estimators

a hausman test for the presence of market microstructure noise in high frequency data abstract we develop tests that help assess whether a high frequency data sample can be treated as reasonably free of market microstructure noise at a given sampling frequency for the purpose of implementing high frequency volatility and other estimators the tests are based on the hausman principle of comparing two estimators one that is efficient but not robust to the deviation being tested and one that is robust but not as efficient we investigate the asymptotic properties of the test statistic in a general nonparametric setting and compare it with several alternatives that are also developed in the paper empirically we find that improvements in stock market liquidity over the past decade have increased the frequency at which simple uncorrected volatility estimators can be safely employed

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

asymptotic error distributions for the euler method for stochastic differential equations we are interested in the rate of convergence of the euler scheme approximation of the solution to a stochastic differential equation driven by a general possibly discontinuous semimartingale and by the asymptotic behavior of the associated normalized error it is well known that for ito s equations the rate is 1 sqrt n we provide a necessary and sufficient condition for this rate to be 1 sqrt n when the driving semimartingale is a continuous martingale or a continuous semimartingale under a mild additional assumption we also prove that in these cases the normalized error processes converge in law the rate can also differ from 1 sqrt n this is the case for instance if the driving process is deterministic or if it is a levy process without a brownian component it is again 1 sqrt n when the driving process is levy with a nonvanishing brownian component but then the normalized error processes converge in law in the finite dimensional sense only while the discretized normalized error processes converge in law in the skorohod sense and the limit is given an explicit form

subsampling realised kernels in a recent paper we have introduced the class of realised kernel estimators of the increments of quadratic variation in the presence of noise we showed that this estimator is consistent and derived its limit distribution under various assumptions on the kernel weights in this paper we extend our analysis looking at the class of subsampled realised kernels and we derive the limit theory for this class of estimators we find that subsampling is highly advantageous for estimators based on discontinuous kernels such as the truncated kernel for kinked kernels such as the bartlett kernel we show that subsampling is impotent in the sense that subsampling has no effect on the asymptotic distribution perhaps surprisingly for the efficientsmooth kernels such as the parzen kernel we show that subsampling is harmful as it increases the asymptotic variance we also study the performance of subsampled realised kernels in simulations and in empirical work

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

minimal bases of matrix pencils algebraic toeplitz structure and geometric properties abstract for a singular matrix pencil sf g the structure of the right rational null space h r is studied and new algebraic and geometric properties and invariants are established the results highlight the nature of the isomorphism between polynomial vectors of h r and real vectors in the right spaces n k r k 0 defined as null spaces of toeplitz matrices of f g the algebraic structure of l r is described by the properties of ordered minimal bases as well as those of new invariants the prime r s modules m i i the geometric structure of h r is defined by the properties of new invariant spaces p i p i r i associated with m i i m the high low and prime spaces as well as some additional invariant spaces defined through the toeplitz representation of ordered minimal bases these new invariant spaces and modules are shown to be naturally defined also on the right spaces n kr and this provides an alternative geometric definition of them which is independent of the original algebraic one it is shown that the construction of an ordered minimal basis is equivalent to a standard linear algebra problem the selection of a right system of generators for n k r k 0 finally certain parametrization issues for the polynomial vectors of h r are considered

bootstrapping realized multivariate volatility measures we study bootstrap methods for statistics that are a function of multivariate high frequency returns such as realized regression coefficients and realized covariances and correlations for these measures of covariation the monte carlo simulation results of barndorff nielsen and shephard 2004 show that finite sample distortions associated with their feasible asymptotic theory approach may arise if sampling is not too frequent this motivates our use of the bootstrap as an alternative tool of inference for covariation measures we consider an i i d bootstrap applied to the vector of returns we show that the finite sample performance of the bootstrap is superior to the existing first order asymptotic theory nevertheless and contrary to the existing results in the bootstrap literature for regression models subject to heteroskedasticity in the error term the edgeworth expansion for the i i d bootstrap that we develop here shows that this method is not second order accurate we argue that this is due to the fact that the conditional mean parameters of realized regression models are heterogeneous under stochastic volatility

testing against constant factor loading matrix with large panel high frequency data abstract in this paper we introduce a nonparametric test against the constancy of the factor loading matrix of a high dimensional continuous time factor model using high frequency data the central limit theorems on the test statistics with and without perturbation are established under the null hypothesis that the factor loading matrix is constant as time evolves the tests perform well in size and power interestingly the test statistic without perturbation converges at a rate that depends not only on the sample size but also on the dimension through the cross sectional dependence of the residual process which is a distinctive feature that contrasts with the low dimensional setting extensive numerical studies including monte carlo simulations and real data analysis validate the performance of our test

assessment of uncertainty in high frequency data the observed asymptotic variance the availability of high frequency financial data has generated a series of estimators based on intra day data improving the quality of large areas of financial econometrics however estimating the standard error of these estimators is often challenging the root of the problem is that traditionally standard errors rely on estimating a theoretically derived asymptotic variance and often this asymptotic variance involves substantially more complex quantities than the original parameter to be estimated standard errors are important they are used to assess the precision of estimators in the form of confidence intervals to create feasible statistics for testing to build forecasting models based on say daily estimates and also to optimize the tuning parameters the contribution of this paper is to provide an alternative and general solution to this problem which we call observed asymptotic variance it is a general nonparametric method for assessing asymptotic variance avar it provides consistent estimators of avar for a broad class of integrated parameters t dt where the spot parameter process can be a general semimartingale with continuous and jump components the observed avar is implemented with the help of a two scales method its construction works well in the presence of microstructure noise and when the observation times are irregular or asynchronous in the multivariate case the methodology is valid for a wide variety of estimators including the standard ones for variance and covariance and also for more complex estimators such as of leverage effects high frequency betas and semivariance

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

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

embedding and asymptotic expansions for martingales the paper develops a way of embedding general martingales in continuous ones in such a way that the quadratic variation of the continuous martingale has conditional cumulants given the original martingale that are explicitly given in terms of optional and predictable variations of the original process bartlett identities for the conditional cumulants are also found a main corollary to these results is the establishment of second and in some cases higher order asymptotic expansions for martingales

efficient estimation of integrated volatility incorporating trading information we consider a setting where market microstructure noise is a parametric function of trading information possibly with a remaining noise component assuming that the remaining noise is op 1 n allowing irregular times and jumps we show that we can estimate the parameters at rate n and propose a volatility estimator which enjoys n convergence rate simulation studies show that our method performs well even with model misspecification and rounding empirical studies demonstrate the practical relevance and advantages of our method furthermore we find that a simple model can account for a high percentage of the total variation in microstructure noise

quasi maximum likelihood estimation of volatility with high frequency data this paper investigates the properties of the well known maximum likelihood estimator in the presence of stochastic volatility and market microstructure noise by extending the classic asymptotic results of quasi maximum likelihood estimation when trying to estimate the integrated volatility and the variance of noise this parametric approach remains consistent efficient and robust as a quasi estimator under misspecified assumptions moreover it shares the model free feature with nonparametric alternatives for instance realized kernels while being advantageous over them in terms of finite sample performance in light of quadratic representation this estimator behaves like an iterative exponential realized kernel asymptotically comparisons with a variety of implementations of the tukey hanning2 kernel are provided using monte carlo simulations and an empirical study with the euro us dollar future illustrates its application in practice

quivers cones and polytopes abstract let q be a quiver without oriented cycles we consider the polytope of flows in q with input these polytopes are closely related to the combinatorial structure of the quiver in particular to its spanning subtrees furthermore we consider a system of cones which turns out to be a fan and can be seen as a base for the family of all flow polytopes for the various inputs finally we present several examples

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

designing realised kernels to measure the ex post variation of equity prices in the presence of noise this paper shows how to use realised kernels to carry out efficient feasible inference on the expost variation of underlying equity prices in the presence of simple models of market frictions the issue is subtle with only estimators which have symmetric weights delivering consistent estimators with mixed gaussian limit theorems the weights can be chosen to achieve the best possible rate of convergence and to have an asymptotic variance which is close to that of the maximum likelihood estimator in the parametric version of this problem realised kernels can also be selected to i be analysed using endogenously spaced data such as that in databases on transactions ii allow for market frictions which are endogenous iii allow for temporally dependent noise the finite sample performance of our estimators is studied using simulation while empirical work illustrates their use in practice

estimating the quadratic covariation matrix for asynchronously observed high frequency stock returns corrupted by additive measurement error this paper studies the estimation problem of the covariance matrices of asset returns in the presence of microstructure noise and asynchronicity between the observations across different assets motivated by malliavin and mancino 2002 2009 we propose a new fourier domain based estimator of multivariate ex post volatility which we call the fourier realized kernel frk an advantage of this approach is that no explicit time alignment is required unlike the time domain based methods widely adopted in the existing literature we derive the large sample properties and establish asymptotic normality of our estimator under some general conditions that allow for both temporal and cross sectional correlations in the measurement error process our results can be viewed as frequency domain extension of the asymptotic theories for the multivariate realized kernel estimator of barndorff nielsen et al 2011 we show in extensive simulations that our method outperforms the time domain estimators when two assets with different liquidity are traded asynchronously

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

mirror symmetry and toric degenerations of partial flag manifolds in this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds f n1 nl n this construction includes our previous mirror construction for complete intersection in grassmannians and the mirror construction of givental for complete flag manifolds the key idea of our construction is a degeneration of f n1 nl n to a certain gorenstein toric fano variety p n1 nl n which has been investigated by gonciulea and lakshmibai we describe a natural small crepant desingularization of p n1 nl n and prove a generalized version of a conjecture of gonciulea and lakshmibai on the singular locus of p n1 nl n mathematisches institut eberhard karls universitat tubingen d 72076 tubingen germany email address batyrev bastau mathematik uni tuebingen de department of mathematics oklahoma state university stillwater ok 74078 usa email address ciocan math okstate edu department of mathematics university of california davis davis ca 95616 usa email address bumsig math ucdavis edu fb 17 mathematik johannes gutenberg universitat mainz d 55099 mainz germany email address straten mathematik uni mainz de

volumes and ehrhart polynomials of flow polytopes the lidskii formula for the type a n root system expresses the volume and ehrhart polynomial of the flow polytope of the complete graph with nonnegative integer netflows in terms of kostant partition functions for every integer polytope the volume is the leading coefficient of the ehrhart polynomial the beauty of the lidskii formula is the revelation that for these polytopes their ehrhart polynomial function can be deduced from their volume function baldoni and vergne generalized lidskii s result for flow polytopes of arbitrary graphs g while their formulas are combinatorial in nature their proofs are based on residue computations in this paper we construct canonical polytopal subdivisions of flow polytopes which we use to prove the baldoni vergne lidskii formulas in contrast with the original computational proof of these formulas our proof reveal their geometry and combinatorics we conclude by exhibiting enumerative properties of the lidskii formulas via our canonical polytopal subdivisions

efficient asymptotic variance reduction when estimating volatility in high frequency data abstract this paper shows how to carry out efficient asymptotic variance reduction when estimating volatility in the presence of stochastic volatility and microstructure noise with the realized kernels rk from barndorff nielsen et al 2008 and the quasi maximum likelihood estimator qmle studied in xiu 2010 to obtain such a reduction we chop the data into b blocks compute the rk or qmle on each block and aggregate the block estimates the ratio of asymptotic variance over the bound of asymptotic efficiency converges as b increases to the ratio in the parametric version of the problem i e 1 0025 in the case of the fastest rk tukey hanning 16 and 1 for the qmle the impact of stochastic sampling times and jump in the price process is examined carefully the finite sample performance of both estimators is investigated in simulations while empirical work illustrates the gain in practice

statistical properties of microstructure noise we study the estimation of joint moments of microstructure noise based on high frequency data the estimation is conducted under a nonparametric setting which allows the underlying price process to have jumps the observation times to be irregularly spaced and the noise to be dependent on the price process and to have diurnal features estimators of arbitrary orders of joint moments are provided for which we establish consistency as well as central limit theorems in particular we provide estimators of autocovariances and autocorrelations of the noise simulation studies demonstrate excellent performance of our estimators in the presence of jumps irregular observation times and even rounding empirical studies reveal moderate positive autocorrelations of microstructure noise for the stocks tested

nonparametric filtering of conditional state price densities abstract this paper studies the use of noisy high frequency data to estimate the time varying state price density implicit in european option prices a dynamic kernel estimator of the conditional pricing function and its derivatives is proposed that can be used for model free risk measurement infill asymptotic theory is derived that applies when the pricing function is either smoothly varying or driven by diffusive state variables trading times and moneyness levels are modeled by marked point processes that capture intraday trading patterns a simulation study investigates the performance of the estimator using a varying plug in bandwidth in various scenarios empirical analysis using s p 500 e mini european option quotes reveals significant time variation at intraday frequencies an application towards delta and minimum variance hedging further illustrates the use of the estimator

a unified approach to volatility estimation in the presence of both rounding and random market microstructure noise abstract widely used volatility estimation methods mainly consider one of the following two simple microstructure noise models random additive noise on log prices or pure rounding errors apparently in real data these two types of noise co exist in this paper we discover a common feature of these two types of noise and propose a unified volatility estimation approach in the presence of both rounding and random noise our data driven method enjoys superior properties in terms of bias and convergence rate we establish feasible central limit theorems and show their superior performance via simulations empirical studies show clear advantages of our method when applied to both stocks data and currency exchange data

ultra high frequency volatility estimation with dependent microstructure noise we analyze the impact of time series dependence in market microstructure noise on the properties of estimators of the integrated volatility of an asset price based on data sampled at frequencies high enough for that noise to be a dominant consideration we show that combining two time scales for that purpose will work even when the noise exhibits time series dependence analyze in that context a refinement of this approach is based on multiple time scales and compare empirically our different estimators to the standard realized volatility

out of sample forecasts of quadratic variation we compare the forecasts of quadratic variation given by the realized volatility rv and the two scales realized volatility tsrv computed from high frequency data in the presence of market microstructure noise under several different dynamics for the volatility process and assumptions on the noise we show that tsrv largely outperforms rv whether looking at bias variance rmse or out of sample forecasting ability an empirical application to all djia stocks confirms the simulation results

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

econometric analysis of realized volatility and its use in estimating stochastic volatility models summary the availability of intraday data on the prices of speculative assets means that we can use quadratic variation like measures of activity in financial markets called realized volatility to study the stochastic properties of returns here under the assumption of a rather general stochastic volatility model we derive the moments and the asymptotic distribution of the realized volatility error the difference between realized volatility and the discretized integrated volatility which we call actual volatility these properties can be used to allow us to estimate the parameters of stochastic volatility models without recourse to the use of simulation intensive methods

computation of matrix fraction descriptions of linear time invariant systems in this paper algorithms are presented for obtaining relatively prime matrix fraction descriptions mfd s of linear time invariant systems orthogonal coordinate transformations are used to reduce a given state space model to one whose state matrix is in block hessenberg form the observability controllability indexes of the system are also obtained in the process a recursive algorithm is then described for obtaining a relatively prime mfd from the block hessenberg representation it is then shown that some simplification can be made in the recursive algorithm by first reducing the block hessenberg form by means of a nonsingular triangular coordinate transformation to a more compact form such as the block frobenius form a permutation of the state variables of the block frobenius form yields a canonical representation similar to the luenberger canonical form some numerical and other properties of the algorithms are discussed and the use of the algorithms is illustrated by numerical examples the numerical performance of the algorithms is also compared with that of the structure algorithm of wolovich and falb

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

estimating covariation epps effect microstructure noise abstract this paper is about how to estimate the integrated covariance x y t of two assets over a fixed time horizon 0 t when the observations of x and y are contaminated and when such noisy observations are at discrete but not synchronized times we show that the usual previous tick covariance estimator is biased and the size of the bias is more pronounced for less liquid assets this is an analytic characterization of the epps effect we also provide the optimal sampling frequency which balances the tradeoff between the bias and various sources of stochastic error terms including nonsynchronous trading microstructure noise and time discretization finally a two scales covariance estimator is provided which simultaneously cancels to first order the epps effect and the effect of microstructure noise the gain is demonstrated in data

a new method for computing a column reduced polynomial matrix a new algorithm is presented for computing a column reduced form of a given full column rank polynomial matrix the method is based on reformulating the problem as a problem of constructing a minimal polynomial basis for the right nullspace of a polynomial matrix closely related to the original one the latter problem can easily be solved in a numerically reliable way three examples illustrating the method are included

a combinatorial model for computing volumes of flow polytopes we introduce new families of combinatorial objects whose enumeration computes volumes of flow polytopes these objects provide an interpretation based on parking functions of baldoni and vergne s generalization of a volume formula originally due to lidskii we recover known flow polytope volume formulas and prove new volume formulas for flow polytopes a highlight of our model is an elegant formula for the flow polytope of a graph we call the caracol graph as by products of our work we uncover a new triangle of numbers that interpolates between catalan numbers and the number of parking functions we prove the log concavity of rows of this triangle along with other sequences derived from volume computations and we introduce a new ehrhart like polynomial for flow polytope volume and conjecture product formulas for the polytopes we consider

anova for diffusions and ito processes ito processes are the most common form of continuous semimartingales and include diffusion processes this paper is concerned with the nonparametric regression relationship between two such ito processes we are interested in the quadratic variation integrated volatility of the residual in this regression over a unit of time such as a day a main conceptual finding is that this quadratic variation can be estimated almost as if the residual process were observed the difference being that there is also a bias which is of the same asymptotic order as the mixed normal error term the proposed methodology anova for diffusions and ito processes can be used to measure the statistical quality of a parametric model and non parametrically the appropriateness of a one regressor model in general on the other hand it also helps quantify and characterize the trading hedging error in the case of financial applications

numerical computation of a coprime factorization of a transfer function matrix abstract a new algorithm is presented for computing a coprime factorization of a transfer function matrix the method is based on reformulating the problem as a problem of constructing a minimal basis of the right nullspace of a matrix pencil b a due to the special structure and rank properties of the pencil obtained after transforming b a to generalized schur form the original problem can be easily solved in a numerically reliable way

continuous record asymptotics for rolling sample variance estimators it is widely known that conditional covariances of asset returns change over time researchers doing empirical work have adopted many strategies for accommodating conditional heteroskedasticity among the popular strategies are a chopping the available data into short blocks of time and assuming homoskedasticity within the blocks b performing one sided rolling regressions in which only data from say the preceding five year period is used to estimate the conditional covariance of returns at a given date and c performing two sided rolling regressions in which covariances are estimated for each date using say five years of lags and five years of leads another model garch amounts to a one sided weighted rolling regression we develop continuous record asymptotic approximations for the measurement error in conditional variances and covariances when using these methods we derive asymptotically optimal window lengths for standard rolling regressions and optimal weights for weighted rolling regressions as an empirical example we estimate volatility on the s p 500 stock index using daily data from 1928 to 1990

column reduction of polynomial matrices a few years ago beelen developed an algorithm to determine a minimal basis for the kernel of a polynomial matrix in this paper we use a modified version of this algorithm to find a column reduced polynomial matrix unimodularly equivalent to a given polynomial matrix