estimating quadratic variation consistently in the presence of endogenous and diurnal measurement error abstract we propose an econometric model that captures the effects of market microstructure on a latent price process in particular we allow for correlation between the measurement error and the return process and we allow the measurement error process to have a diurnal heteroskedasticity we propose a modification of the tsrv estimator of quadratic variation we show that this estimator is consistent with a rate of convergence that depends on the size of the measurement error but is no worse than n 1 6 we investigate in simulation experiments the finite sample performance of various proposed implementations

subsampling high frequency data the main contribution of this paper is to propose a novel way of conducting inference for an important general class of estimators that includes many estimators of integrated volatility a subsampling scheme is introduced that consistently estimates the asymptotic variance for an estimator thereby facilitating inference and the construction of valid confidence intervals the new method does not rely on the exact form of the asymptotic variance which is useful when the latter is of complicated form the method is applied to the volatility estimator of ait sahalia et al 2011 in the presence of autocorrelated and heteroscedastic market microstructure noise

fokker planck equation for a square lattice of coupled anharmonic oscillators and the two dimensional ising model the authors calculate the lowest non zero eigenvalue of the fokker planck equation for an infinite square lattice of coupled overdamped anharmonic oscillators bistable potentials through a variational calculation it generalises previous work for only one oscillator in a non trivial way they reduce that calculation to that of a suitable ratio of partition functions for the 2d ising model for which an exact analytical expression is obtained numerical results are also given and discussed

violation of universality induced by critical dynamics the mapping of critical dynamics onto a static d 1 dimensional system is analysed in the large n limit at criticality lifshitz tricritical behaviour with upper critical dimensionality d 4 occurs in the neighbourhood of an ordinary lifshitz point with d 6 the violation of universality is a consequence of the very special path along which the critical point is approached

field theoretic study of the nonlinear fokker planck equation a new field theoretic formulation of the fokker planck approach to non equilibrium statistical mechanics is presented starting with the nonlinear functional fokker planck equation a new generating functional is derived no use of auxiliary conjugate fields or response functions is needed the feynman rules are deduced and the renormalisation of the theory is carried out finally the renormalisation group equation is solved and scaling laws and critical exponents are calculated which are in good agreement with previous results obtained through different formalisms

a bias corrected estimator of the covariation matrix of multiple security prices when both microstructure effects and sampling durations are persistent and endogenous i propose a bias corrected non parametric estimator of the covariation matrix of log security prices designed as a convex combination of two realized kernels the estimator is simple but possesses desirable statistical properties including consistency asymptotic normality and the parametric rate of convergence in the presence of persistent diurnally heteroskedastic and endogenous microstructure effects it is robust to the asynchronous trading of multiple securities with persistent and endogenous refresh time durations i also prove the consistency of a subsampling based estimator of the asymptotic covariance matrix of the proposed estimator in simulations the non linear functions of the proposed estimator exhibit smaller bias than those based on a realized kernel while only slightly increasing the variance thereby the proposed estimator reduces the mean squared error

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

homotopy algorithm for symmetric eigenvalue problems the homotopy method can be used to solve eigenvalue eigenvector problems the purpose of this paper is to report the numerical experience of the homotopy method of computing eigenpairs for real symmetric tridiagonal matrices together with a couple of new theoretical results in practice it is rerely of any interest to compute all the eigenvalues the homotopy method having the order preserving property can provide any specific eigenvalue without calculating any other eigenvalues besides this advantage we note that the homotopy algorithm is to a large degree a parallel algorithm numerical experimentation shows that the homotopy method can be very efficient especially for graded matrices

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

a new shift of the ql algorithm for irreducible symmetric tridiagonal matrices a new shift in the ql algorithm for symmetric tridiagonal matrices is described the shift is a combination of the rayleigh quotient shift and wilkinson s shift it is shown that ql is globally convergent with this shift and that the asymptotic rate is always cubic

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

estimating the quadratic covariation matrix from noisy observations local method of moments and efficiency an efficient estimator is constructed for the quadratic covariation or integrated covolatility matrix of a multivariate continuous martingale based on noisy and non synchronous observations under high frequency asymptotics our approach relies on an asymptotically equivalent continuous time observation model where a local generalised method of moments in the spectral domain turns out to be optimal asymptotic semiparametric efficiency is established in the cramer rao sense main findings are that non synchronicity of observation times has no impact on the asymptotics and that major efficiency gains are possible under correlation simulations illustrate the finite sample behaviour

econometric analysis of realized covariation high frequency based covariance regression and correlation in financial economics this paper analyses multivariate high frequency financial data using realized covariation we provide a new asymptotic distribution theory for standard methods such as regression correlation analysis and covariance it will be based on a fixed interval of time e g a day or week allowing the number of high frequency returns during this period to go to infinity our analysis allows us to study how high frequency correlations regressions and covariances change through time in particular we provide confidence intervals for each of these quantities

empirical bias bandwidths for local polynomial nonparametric regression and density estimation abstract a data based local bandwidth selector is proposed for nonparametric regression by local fitting of polynomials the estimator called the empirical bias bandwidth selector ebbs is rather simple and easily allows multivariate predictor variables and estimation of any order derivative of the regression function ebbs minimizes an estimate of mean squared error consisting of a squared bias term plus a variance term the variance term used is exact not asymptotic though it involves the conditional variance of the response given the predictors that must be estimated the bias term is estimated empirically not from an asymptotic expression thus ebbs is similar to the double smoothing approach of hardle hall and marron and a local bandwidth selector of schucany but is developed here for a far wider class of estimation problems than what those authors considered ebbs is tested on simulated data and its performance seems quite satisfactory local polynomial smoothing of a histogram is a high

an algorithm for finding generalized eigenpairs of a symmetric definite matrix pencil abstract a new method for finding eigenpairs of any symmetric definite matrix pencil is proposed it is a modification of the well known rayleigh quotient iteration it circumvents a drawback of that method i e its nonconvergence or slow convergence for poor initial vectors the global convergence and cubic convergence rate of the new method are proved its efficiency is also demonstrated by numerical results

estimating the integrated volatility with tick observations abstract we develop a volatility estimator that can be directly applied to tick by tick data more specifically we consider a model that allows for i irregular observation times that can be endogenous ii dependent noise that can have diurnal features and be dependent on the latent price process and iii jumps in the latent price process we show that our estimator yields consistent estimates and enjoys the optimal rate of convergence simulation as well as empirical studies demonstrate favorable properties of our proposed estimator

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

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

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

knowing factors or factor loadings or neither evaluating estimators of large covariance matrices with noisy and asynchronous data abstract we investigate estimators of factor model based large covariance and precision matrices using high frequency data which are asynchronous and potentially contaminated by the market microstructure noise our estimation strategies rely on the pre averaging method with refresh time to solve the microstructure problems while using three different specifications of factor models with a variety of thresholding methods respectively to battle the curse of dimensionality to estimate a factor model we either adopt the time series regression tsr to recover loadings if factors are known or use the cross sectional regression csr to recover factors from known loadings or use the principal component analysis pca if neither factors nor their loadings are assumed known we compare the convergence rates in these scenarios using the joint in fill and increasing dimensionality asymptotics to evaluate the empirical trade off between robustness to model misspecification and statistical efficiency among all 30 combinations of estimation strategies we run a horse race on the out of sample portfolio allocation with dow jones 30 s p 100 and s p 500 index constituents respectively and find the pre averaging based strategy using tsr or pca with location thresholding dominates especially over the subsampling based alternatives

asymptotic equivalence for inference on the volatility from noisy observations we consider discrete time observations of a continuous martingale under measurement error this serves as a fundamental model for high frequency data in finance where an efficient price process is observed under microstructure noise it is shown that this nonparametric model is in le cam s sense asymptotically equivalent to a gaussian shift experiment in terms of the square root of the volatility function and a nonstandard noise level as an application new rate optimal estimators of the volatility function and simple efficient estimators of the integrated volatility are constructed

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

a tale of two time scales it is a common practice in finance to estimate volatility from the sum of frequently sampled squared returns however market microstructure poses challenges to this estimation approach as evidenced by recent empirical studies in finance the present work attempts to lay out theoretical grounds that reconcile continuous time modeling and discrete time samples we propose an estimation approach that takes advantage of the rich sources in tick by tick data while preserving the continuous time assumption on the underlying returns under our framework it becomes clear why and where the usual volatility estimator fails when the returns are sampled at the highest frequencies if the noise is asymptotically small our work provides a way of finding the optimal sampling frequency a better approach the two scales estimator works for any size of the noise