essential spectrum of a periodic waveguide with non periodic perturbation abstract we consider the spectral dirichlet laplacian problem on a domain which is formed from a periodic waveguide perturbed by non compact non periodic changes of geometry we show that the domain perturbation causes an addition to the essential spectrum which consists of isolated points belonging to the discrete spectrum of a model problem this model problem is posed on a domain which is just a compact perturbation of we discuss the position of the new spectral components in relation to the essential spectrum of the problem in

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

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

functional data analysis for volatility abstract we introduce a functional volatility process for modeling volatility trajectories for high frequency observations in financial markets and describe functional representations and data based recovery of the process from repeated observations a study of its asymptotic properties as the frequency of observed trades increases is complemented by simulations and an application to the analysis of intra day volatility patterns of the s p 500 index the proposed volatility model is found to be useful to identify recurring patterns of volatility and for successful prediction of future volatility through the application of functional regression and prediction techniques

bayesian vector autoregressions with stochastic volatility this paper proposes a bayesian approach to a vector autoregression with stochastic volatility where the multiplicative evolution of the precision matrix is driven by a multivariate beta variate exact updating formulas are given to the nonlinear filtering of the precision matrix estimation of the autoregressive parameters requires numerical methods an importance sampling based approach is explained here

the realized empirical distribution function of stochastic variance with application to goodness of fit testing abstract we propose a nonparametric estimator of the empirical distribution function edf of the latent spot variance of the log price of a financial asset we show that over a fixed time span our realized edf or redf inferred from noisy high frequency data is consistent as the mesh of the observation grid goes to zero in a double asymptotic framework with time also increasing to infinity the redf converges to the cumulative distribution function of volatility if it exists we exploit these results to construct some new goodness of fit tests for stochastic volatility models in a monte carlo study the redf is found to be accurate over the entire support of volatility this leads to goodness of fit tests that are both correctly sized and relatively powerful against common alternatives in an empirical application we recover the redf from stock market high frequency data we inspect the goodness of fit of several two parameter marginal distributions that are inherent in standard stochastic volatility models the inverse gaussian offers the best overall description of random equity variation but the fit is less than perfect this suggests an extra parameter as available in e g the generalized inverse gaussian is required to model stochastic variance

threshold estimation of markov models with jumps and interest rate modeling we reconstruct the level dependent diffusion coefficient of a univariate semimartingale with jumps which is observed discretely the consistency and asymptotic normality of our estimator are provided in the presence of both finite and infinite activity finite variation jumps our results rely on kernel estimation using the properties of the local time of the data generating process and the fact that it is possible to disentangle the discontinuous part of the state variable through those squared increments between observations not exceeding a suitable threshold function we also reconstruct the drift and the jump intensity coefficients when they are level dependent and jumps have finite activity through consistent and asymptotically normal estimators simulated experiments show that the newly proposed estimators perform better in finite samples than alternative estimators and this allows us to reexamine the estimation of a univariate model for the short term interest rate for which we find fewer jumps and more variance due to the diffusion part than previous studies

spectra of open waveguides in periodic media abstract we study the essential spectra of formally self adjoint elliptic systems on doubly periodic planar domains perturbed by a semi infinite periodic row of foreign inclusions we show that the essential spectrum of the problem consists of the essential spectrum of the purely periodic problem and another component which is the union of the discrete spectra of model problems in the infinite perturbation strip these model problems arise by an application of the partial floquet bloch gelfand transform

modeling maxima with autoregressive conditional frechet model abstract this paper introduces a novel dynamic generalized extreme value gev framework for modeling the time varying behavior of maxima in financial time series specifically an autoregressive conditional frechet acf model is proposed in which the maxima are modeled by a frechet distribution with time varying scale parameter volatility and shape parameter tail index conditioned on past information the acf provides a direct and accurate modeling of the time varying behavior of maxima and furthermore offers a new angle to study the tail risk dynamics in financial markets probabilistic properties of acf are studied and a maximum likelihood estimator is used for model estimation with its statistical properties investigated simulations show the flexibility of acf and confirm the reliability of its estimators two real data examples on cross sectional stock returns and high frequency foreign exchange returns are used to demonstrate the acf modeling approach where significant improvement over the static gev has been observed for market tail risk monitoring and conditional var estimation empirical result of acf is consistent with the findings of the dynamic peak over threshold pot literature that the tail index of financial markets varies through time

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

likelihood analysis of non gaussian measurement time series summary in this paper we provide methods for estimating non gaussian time series models these techniques rely on markov chain monte carlo to carry out simulation smoothing and bayesian posterior analysis of parameters and on importance sampling to estimate the likelihood function for classical inference the time series structure of the models is used to ensure that our simulation algorithms are efficient

stochastic multifactor modeling of spot electricity prices in this paper a stochastic multifactor model is proposed for modeling of the daily spot market electricity prices stochastic part of the model is composed of three jump processes and a brownian motion where two of the jump processes are assumed to be mean reverting with different mean reversion rates the multistep algorithm proposed for model estimation utilizes an iterative threshold function constructed on garch 1 1 volatility estimates in separation of the jumps the factor model is applied to turkish day ahead electricity market in order to evaluate the performance of the proposed multifactor model estimation results are also compared to the results acquired by application of mean reverting jump diffusion model of cartea and figueroa 2005 and markov regime switching model of janczura and weron 2010 to the same data set

modelling price dynamics a hybrid truncated levy flight garch approach abstract arch and garch stochastic processes are widely used in finance and are generally accepted as good approximations when modelling the price dynamics with gaussian conditional probability it can be seen that certain aspects of the empirical data for asset price changes seems to more closely fit a truncated levy flight or garch model but each with individual shortfalls in this paper therefore we combine the garch process with a conditional truncated levy distribution in order to build a hybrid model that most notably describes the price change and associated volatility probability density distributions and scaling behaviour over different time horizons

nonparametric inference of discretely sampled stable levy processes abstract we study nonparametric inference of stochastic models driven by stable levy processes we introduce a nonparametric estimator of the stable index that achieves the parametric n rate of convergence for the volatility function due to the heavy tailedness the classical least squares method is not applicable we then propose a nonparametric least absolute deviation or median quantile estimator and study its asymptotic behavior including asymptotic normality and maximal deviations by establishing a representation of bahadur kiefer type the result is applied to several major foreign exchange rates

classification and conformal symmetry in three dimensional space times three dimensional manifolds admitting lorentz metrics are studied the first part of the paper gives a classification of the ricci and curvature tensors and also of the conformal schouten cotton york tensor the second part of the paper investigates killing and conformal symmetry and also the nature of the zeros of the associated vector fields the maximum dimension of the killing and conformal algebras is calculated a theorem regarding the reduction of the conformal algebra to a killing algebra of a conformally related metric is given

on the extremes of a class of non linear processes with heavy tailed innovations the aim of this paper is to look at the limiting form of certain empirical point processes induced by a particular class of non linear processes generated by heavy tailed innovations such asymptotic results will be highly useful in obtaining the weak limiting behavior of various functionals of the underlying process including the asymptotic distribution of upper and lower order statistics in particular we investigate the maximum limiting distribution and its corresponding extremal index the results are applied to the study of the extremal properties of bilinear processes

on the invariance of the lagrange multiplier test with respect to certain changes in the alternative hypothesis this paper examines some implications of the observation that the same lagrange multiplier test is sometimes appropriate for quite different alternative hypotheses a characterization of the class of such alternatives is developed which suggests a simple approach to testing for misspecification and the consequences for finite sample power properties are examined by monte carlo experiments

periodic manifolds with spectral gaps abstract we investigate spectral properties of the laplace operator on a class of non compact riemannian manifolds for a given number n we construct periodic manifolds such that the essential spectrum of the corresponding laplacian has at least n open gaps we use two different methods first we construct a periodic manifold starting from an infinite number of copies of a compact manifold connected by small cylinders in the second construction we begin with a periodic manifold which will be conformally deformed in both constructions a decoupling of the different period cells is responsible for the gaps

value at risk estimation with wavelet based extreme value theory evidence from emerging markets abstract this paper introduces wavelet based extreme value theory evt for univariate value at risk estimation wavelets and evt are combined for volatility forecasting to estimate a hybrid model in the first stage wavelets are used as a threshold in generalized pareto distribution and in the second stage evt is applied with a wavelet based threshold this new model is applied to two major emerging stock markets the istanbul stock exchange ise and the budapest stock exchange bux the relative performance of wavelet based evt is benchmarked against the riskmetrics ewma arma garch generalized pareto distribution and conditional generalized pareto distribution models the empirical results show that the wavelet based extreme value theory increases predictive performance of financial forecasting according to number of violations and tail loss tests the superior forecasting performance of the wavelet based evt model is also consistent with basel ii requirements and this new model can be used by financial institutions as well

estimating spot volatility with high frequency financial data we construct a spot volatility estimator for high frequency financial data which contain market microstructure noise we prove consistency and derive the asymptotic distribution of the estimator a data driven method is proposed to select the scale parameter and the bandwidth parameter in the estimator in monte carlo simulations we compare the finite sample performance of our estimator with some existing estimators empirical examples are given to illustrate the potential applications of the estimator

a top bottom price approach to understanding financial fluctuations abstract the presence of sequences of top and bottom tb events in financial series is investigated for the purpose of characterizing such switching points they clearly mark a change in the trend of rising or falling prices of assets to the opposite tendency are of crucial importance for the players decision and also for the market stability previous attempts to characterize switching points have been based on the behavior of the volatility and on the definition of microtrends the approach used herein is based on the smoothing of the original data with a gaussian kernel the events are identified by the magnitude of the difference of the extreme prices by the time lag between the corresponding events waiting time and by the time interval between events with a minimal magnitude return time results from the analysis of the inter day dow jones industrial average index djia from 1928 to 2011 are discussed q gaussian functions with power law tails are found to provide a very accurate description of a class of measures obtained from the series statistics

impact of foreign exchange rate on oil companies risk in stock market during the recent years the importance of effective risk management has become extremely crucial value at risk var is a standard downside measure to explain the behavior of financial series as the garch models have been successfully applied in modeling the volatility structure of securities and other financial assets the var of garch models has turned into an important quantity to study in addition to garch family models the markov regime switching garch mrs garch models are a widely used approach to model financial volatility with potential structural breaks in this paper we evaluated the performance of the mrs garch models with the traditional arma garch family models for estimating var of the stock return of operating companies in the oil industry srocoi in irans stock market under the normal student t and ged distributions our findings showed that the mrs garch models outperform the arma garch family models to capture the characteristics of the srocois volatility additionally we evaluated impact of the foreign exchange rate fluctuations on the var of srocois that were calculated from the optimal models under two regimes usingardl model the results further demonstrated that foreign exchange rate fluctuations have significant and different impacts on the var of srocois across regimes we modeled srocois in irans stock market using mrs garch and arma garch family models mrs garch models under student t and ged distributions have better performance to capture srocois we estimated in sample and out of sample var of srocois from optima models we evaluated impact of the foreign exchange rate fluctuations on the var of srocois foreign exchange rate significantly effect on the var of srocois in different regimes

affine conformal vectors in space time all space times admitting a proper affine conformal vector acv are found by using a theorem of hall and da costa it is shown that such space times either i admit a covariantly constant vector timelike spacelike or null and the acv is the sum of a proper affine vector and a conformal killing vector or ii the space time is 2 2 decomposable in which case it is shown that no acv can exist unless the space time decomposes further furthermore it is proved that all space times admitting an acv and a null covariantly constant vector which are necessarily generalized pp wave space times must have ricci tensor of segre type 2 1 1 it follows that among space times admitting proper acv the einstein static universe is the only perfect fluid space time there are no non null einstein maxwell space times and only the pp wave space times are representative of null einstein maxwell solutions otherwise the space times can represent anisotropic fluids and viscous heat conducting fluids but only with restricted equations of state in each case

non gaussian state space modeling of nonstationary time series abstract a non gaussian state space approach to the modeling of nonstationary time series is shown the model is expressed in state space form where the system noise and the observational noise are not necessarily gaussian recursive formulas of prediction filtering and smoothing for the state estimation and identification of the non gaussian state space model are given also given is a numerical method based on piecewise linear approximation to the density functions for realizing these formulas significant merits of non gaussian modeling and the wide range of applicability of the method are illustrated by some numerical examples a typical application of this non gaussian modeling is the smoothing of a time series that has mean value function with both abrupt and gradual changes simple gaussian state space modeling is not adequate for this situation here the model with small system noise variance cannot detect jump whereas the one with large system noise variance yields unfavorable wiggle to work

double warped space times an invariant characterization of double warped space times is given in terms of newman penrose formalism and a classification scheme is proposed a detailed study of the conformal algebra of these space times is also carried out and some remarks are made on certain classes of exact solutions

asymptotic filtering theory for univariate arch models this paper builds on this earlier work by deriving the asymptotic distribution of the measurement error this allows us to approximate the measurement accuracy of arch conditional variance estimates and compare the efficiency achieved by different arch models we are also able to characterize the relative importance of different kinds of misspecification for example we show that misspecifying conditional means adds only trivially at least asymptotically to measurement error while other factors for example capturing the leverage effect accommodating thick tailed residuals and correctly modelling the variability of the conditional variance process are potentially much more important third we are able to characterize a class of asymptotically optimal arch conditional variance estimates

estimating stochastic volatility diffusion using conditional moments of integrated volatility we exploit the distributional information contained in high frequency intraday data in constructing a simple conditional moment estimator for stochastic volatility diffusions the estimator is based on the analytical solutions of the first two conditional moments for the latent integrated volatility the realization of which is effectively approximated by the sum of the squared high frequency increments of the process our simulation evidence indicates that the resulting gmm estimator is highly reliable and accurate our empirical implementation based on high frequency five minute foreign exchange returns suggests the presence of multiple latent stochastic volatility factors and possible jumps

fractionally integrated generalized autoregressive conditional heteroskedasticity abstract the new class of fractionally integrated generalized autoregressive conditionally heteroskedastic figarch processes is introduced the conditional variance of the process implies a slow hyperbolic rate of decay for the influence of lagged squared innovations unlike i d processes for the mean maximum likelihood estimates mle of the figarch parameters are argued to be t 1 2 consistent the small sample behavior of an approximate mle procedure is assessed through a simulation study which also documents how the estimation of a standard garch model tends to produce integrated or igarch like estimates an empirical example with daily deutschmark u s dollar exchange rates illustrates the practical relevance of the new figarch specification

integrated variance forecasting model based vs reduced form abstract this paper compares model based and reduced form forecasts of financial volatility when high frequency return data are available we derived exact formulas for the forecast errors and analyzed the contribution of the wrong data modeling and errors in forecast inputs the comparison is made for feasible forecasts i e we assumed that the true data generating process latent states and parameters are unknown as an illustration the same comparison is carried out empirically for spot 5 min returns of dm usd exchange rates it is shown that the comparison between feasible reduced form and model based forecasts is not always in favor of the latter in contrast to their infeasible versions the reduced form approach is generally better for long horizon forecasting and for short horizon forecasting in the presence of microstructure noise

smoothing and interpolation with the state space model abstract a new result dealing with smoothing and interpolation in the state space model is developed and explored the result simplifies the derivation of existing smoothing algorithms and provides alternate forms that have analytic and practical computing advantages the connection to signal extraction and interpolation is explored and diffuse specifications are considered

data based ranking of realised volatility estimators abstract this paper presents new methods for comparing the accuracy of estimators of the quadratic variation of a price process i provide conditions under which the relative accuracy of competing estimators can be consistently estimated as t and show that forecast evaluation tests may be adapted to the problem of ranking these estimators the proposed methods avoid making specific assumptions about microstructure noise and facilitate comparisons of estimators that would be difficult using methods from the extant literature such as those based on different sampling schemes an application to high frequency ibm data between 1996 and 2007 illustrates the new methods

modeling and inference with υ spherical distributions abstract a new class of continuous multivariate distributions on n is proposed we define these so called spherical distributions through properties of the density function in a location scale context we derive conditions for properness of spherical distributions and discuss how to generate them in practice the name spherical is motivated by the fact that these distributions generalize the classes of spherical when is the l 2 norm and l q spherical when is the l q norm distributions isodensity sets are still always situated around the location parameter but exchangeability and axial symmetry are no longer imposed as is illustrated in some examples as an important special case we define a class of distributions suggested by independent sampling from a generalization of exponential power distributions this allows us to model skewness interestingly all the robustness results found previously for spherical and l q spherical models carry over directly to spherical mo

occupation density estimation for noisy high frequency data abstract this paper studies the nonparametric estimation of occupation densities for semimartingale processes observed with noise as leading examples we consider the stochastic volatility of a latent efficient price process the volatility of the latent noise that separates the efficient price from the actually observed price and nonlinear transformations of these processes our estimation methods are decidedly nonparametric and consist of two steps the estimation of the spot price and noise volatility processes based on pre averaging techniques and in fill asymptotic arguments followed by a kernel type estimation of the occupation densities our spot volatility estimates attain the optimal rate of convergence and are robust to leverage effects price and volatility jumps general forms of serial dependence in the noise and random irregular sampling the convergence rates of our occupation density estimates are directly related to that of the estimated spot volatilities and the smoothness of the true occupation densities an empirical application involving high frequency equity data illustrates the usefulness of the new methods in illuminating time varying risks market liquidity and informational asymmetries across time and assets

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

monte carlo inference in econometric models with symmetric stable disturbances the paper develops markov chain monte carlo methods to perform exact posterior analysis in models with symmetric stable paretian disturbances it is shown how posterior moments and marginal densities of functions of parameters can be computed methodically by combining a gibbs sampler with metropolis independence chains the new method is shown to perform satisfactorily in constructed examples the method is also applied to a set of monthly real exchange rates and the question of difference versus trend stationarity is taken up jointly with the problem of inference about the parameter of the stable distribution

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

interval estimation of value at risk based on garch models with heavy tailed innovations arch and garch models are widely used to model financial market volatilities in risk management applications considering a garch model with heavy tailed innovations we characterize the limiting distribution of an estimator of the conditional value at risk var which corresponds to the extremal quantile of the conditional distribution of the garch process we propose two methods the normal approximation method and the data tilting method for constructing confidence intervals for the conditional var estimator and assess their accuracies by simulation studies finally we apply the proposed approach to an energy market data set

adaptive estimation in time series models in a framework particularly suited for many time series models we obtain a lan result under quite natural and economical conditions this enables us to construct adaptive estimators for part of the euclidean parameter in these semiparametric models special attention is directed to group models in time series with the important subclass of models with time varying location and scale our set up is confronted with the existing literature and as examples we reconsider linear regression and arma tar and arch models

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

managing distribution changes in time series prediction when a problem is modeled statistically a single distribution model is usually postulated that is assumed to be valid for the entire space nonetheless this practice may be somewhat unrealistic in certain application areas in which the conditions of the process that generates the data may change as far as we are aware however no techniques have been developed to tackle this problem this article proposes a technique for modeling and predicting this change in time series with a view to improving estimates and predictions the technique is applied among other models to the hypernormal distribution recently proposed when tested on real data from a range of stock market indices the technique produces better results that when a single distribution model is assumed to be valid for the entire period of time studied moreover when a global model is postulated it is highly recommended to select the hypernormal distribution parameter in the same likelihood maximization process

maximum entropy autoregressive conditional heteroskedasticity model abstract in many applications it has been found that the autoregressive conditional heteroskedasticity arch model under the conditional normal or student s t distributions are not general enough to account for the excess kurtosis in the data moreover asymmetry in the financial data is rarely modeled in a systematic way in this paper we suggest a general density function based on the maximum entropy me approach that takes account of asymmetry excess kurtosis and also of high peakedness the me principle is based on the efficient use of available information and as is well known many of the standard family of distributions can be derived from the me approach we demonstrate how we can extract information functional from the data in the form of moment functions we also propose a test procedure for selecting appropriate moment functions our procedure is illustrated with an application to the nyse stock returns the empirical results reveal that the me approach with a fewer moment functions leads to a model that captures the stylized facts quite effectively

variance function estimation abstract heteroscedastic regression models are used in fields including economics engineering and the biological and physical sciences often the heteroscedasticity is modeled as a function of the covariates or the regression and other structural parameters standard asymptotic theory implies that how one estimates the variance function in particular the structural parameters has no effect on the first order properties of the regression parameter estimates there is evidence however both in practice and higher order theory to suggest that how one estimates the variance function does matter further in some settings estimation of the variance function is of independent interest or plays an important role in estimation of other quantities in this article we study variance function estimation in a unified way focusing on common methods proposed in the statistical and other literature to make both general observations and compare different estimation schemes we show that there are significant di

filtering and forecasting with misspecified arch models i getting the right variance with the wrong model this paper investigates the properties of the conditional covariance estimates generated by a misspecified arch model for example suppose that we observe a diffusion process at discrete time intervals of length h for each h we use a garch 1 1 model to estimate the instantaneous conditional covariance matrix of the diffusion under mild regularity conditions the difference between these conditional covariance estimates and the true conditional covariance converges to zero in probability as h 0 many other arch models for example exponential arch have similar consistency properties this may well account for the success of arch models in short term forecasting using high frequency data since even misspecified arch models can produce good estimates of volatility

asymptotically optimal smoothing with arch models suppose an observed time series is generated by a stochastic volatility model i e there is an unobservable state variable controlling the volatility of the innovations in the series as shown by nelson 1992 and nelson and foster 1994 a misspecified arch model will often be able to consistently as a continuous time limit is approached estimate the unobserved volatility process using information in the lagged residuals this paper shows how to more efficiently estimate such a volatility process using information in both lagged and led residuals in particular this paper expands the optimal filtering results of nelson and foster 1994 and nelson 1994 to smoothing and to filtering with a random initial condition

the distribution of realized exchange rate volatility using high frequency data on deutschemark and yen returns against the dollar we construct model free estimates of daily exchange rate volatility and correlation that cover an entire decade our estimates termed realized volatilities and correlations are not only model free but also approximately free of measurement error under general conditions which we discuss in detail hence for practical purposes we may treat the exchange rate volatilities and correlations as observed rather than latent we do so and we characterize their joint distribution both unconditionally and conditionally noteworthy results include a simple normality inducing volatility transformation high contemporaneous correlation across volatilities high correlation between correlation and volatilities pronounced and persistent dynamics in volatilities and correlations evidence of long memory dynamics in volatilities and correlations and remarkably precise scaling laws under temporal aggregation

curvature structure in general relativity this paper investigates the extent to which the curvature structure of space time determines the metric stucture it continues the work of earlier papers by prescribing the curvature structure and the curvature covariant derivatives up to certain orders it is shown that with the exception of the so called generalized pp waves the curvature and its first and second covariant derivatives essentially determine the metric up to coordinate transformations

arch models as diffusion approximations abstract this paper investigates the convergence of stochastic difference equations e g arch to stochastic differential equations as the length of the discrete time intervals between observations goes to zero these results are applied to the garch 1 1 model of bollerslev 1986 and to the ar 1 exponential arch model of nelson 1989 in their continuous time limits the conditional variance processes in these models have stationary distributions that are inverted gamma and lognormal respectively in addition a class of diffusion approximations based on the exponential arch model is developed

time evolution of stochastic processes with correlations in the variance stability in power law tails of distributions abstract we model the time series of the s p500 index by a combined process the ar garch process where ar denotes the autoregressive process which we use to account for the short range correlations in the index changes and garch denotes the generalized autoregressive conditional heteroskedastic process which takes into account the long range correlations in the variance we study the ar garch process with an initial distribution of truncated levy form we find that this process generates a new probability distribution with a crossover from a levy stable power law to a power law with an exponent outside the levy range beyond the truncation cutoff we analyze the sum of n variables of the ar garch process and find that due to the correlations the ar garch process generates a probability distribution which exhibits stable behavior in the tails for a broad range of values n a feature which is observed in the probability distribution of the s p500 index we find that this power law stability depends on the characteristic scale in the correlations we also find that inclusion of short range correlations through the ar process is needed to obtain convergence to a limiting gaussian distribution for large n as observed in the data

exponential family state space models based on a conjugate latent process summary this paper introduces non linear and non gaussian state space models with analytic updating recurnions for filtering and prediction this new class of models involves some well known results in the theory of exponential models and of exponential dispersion models and the latent process is defined in such a way that both the filtering and the prediction distributions turn out to be conjugate to the observation distribution at each step the corresponding analytic and inferential properties are investigated and some simple examples are presented

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

the realized laplace transform of volatility we introduce and derive the asymptotic behavior of a new measure constructed from high frequency data which we call the realized laplace transform of volatility the statistic provides a nonparametric estimate for the empirical laplace transform function of the latent stochastic volatility process over a given interval of time and is robust to the presence of jumps in the price process with a long span of data that is under joint long span and infill asymptotics the statistic can be used to construct a nonparametric estimate of the volatility laplace transform as well as of the integrated joint laplace transform of volatility over different points of time we derive feasible functional limit theorems for our statistic both under fixed span and infill asymptotics as well as under joint long span and infill asymptotics which allow us to quantify the precision in estimation under both sampling schemes

forecasting s p 100 volatility the incremental information content of implied volatilities and high frequency index returns the information content of implied volatilities and intraday returns is compared in the context of forecasting index volatility over horizons from 1 to 20 days forecasts of two measures of realized volatility are obtained after estimating arch models using daily index returns daily observations of the vix index of implied volatility and sums of squares of 5 min index returns the in sample estimates show that nearly all relevant information is provided by the vix index and hence there is not much incremental information in high frequency index returns for out of sample forecasting the vix index provides the most accurate forecasts for all forecast horizons and performance measures considered the evidence for incremental forecasting information in intraday returns is insignificant

forecasting volatility of ssec in chinese stock market using multifractal analysis abstract in this paper taking about 7 years high frequency data of the shanghai stock exchange composite index ssec as an example we propose a daily volatility measure based on the multifractal spectrum of the high frequency price variability within a trading day an arfima model is used to depict the dynamics of this multifractal volatility mfv measures the one day ahead volatility forecasting performances of the mfv model and some other existing volatility models such as the realized volatility model stochastic volatility model and garch are evaluated by the superior prediction ability spa test the empirical results show that under several loss functions the mfv model obtains the best forecasting accuracy

estimating semiparametric arch models by kernel smoothing methods we investigate a class of semiparametric arch models that includes as a special case the partially nonparametric pnp model introduced by engle and ng 1993 and which allows for both flexible dynamics and flexible function form with regard to the news impact function we show that the functional part of the model satisfies a type ii linear integral equation and give simple conditions under which there is a unique solution we propose an estimation method that is based on kernel smoothing and profiled likelihood we establish the distribution theory of the parametric components and the pointwise distribution of the nonparametric component of the model we also discuss efficiency of both the parametric part and the nonparametric part we investigate the performance of our procedures on simulated data and on a sample of s p500 index returns we find evidence of asymmetric news impact functions consistent with the parametric analysis

conformal vector fields in general relativity a general discussion of conformal vector fields in space times is given amongst the topics considered are the maximum dimension of the conformal algebra for space times that are not conformally flat the nature of conformal isotropies and a new approach to the theorem of bilyalov and defrise carter concerning the reduction of the conformal algebra to a killing or homothetic algebra some deficiencies in the original statements of this theorem are discussed with reference to a general class of counterexamples and corrected the proof offered is geometrical in nature and has the advantage of displaying some of the more general features and properties of conformal vector fields and the ways in which they can differ from killing vector fields

quasi maximum likelihood estimation for conditional quantiles in this paper we derive the asymptotic distribution of a new class of quasi maximum likelihood estimators qmle based on a tick exponential family of densities we show that the tick exponential assumption is a necessary and sufficient condition for a qmle to be consistent for the parameters of a correctly specified model of a given conditional quantile hence the role of this family of densities in the conditional quantile estimation is analog to the role of the linear exponential family in the conditional mean estimation the tick exponential qmles are shown to be asymptotically normal with an asymptotic covariance matrix that has a novel form not seen in earlier work and which accounts for possible model misspecification for practical purposes we show that the maximization of the tick exponential quasi log likelihood can conveniently be carried out by using standard gradient based optimization techniques more importantly we provide a consistent estimator for the asymptotic covariance matrix based on the scores of the log likelihood which allows us to compute the conditional quantile confidence intervals

modelling high frequency economic time series abstract the minute by minute move of the hang seng index hsi data over a 4 yr period is analysed and shown to possess similar statistical features as those of other markets based on a mathematical theorem pope ching phys fluids a 5 1993 1529 we derive an analytic form for the probability distribution function pdf of index moves from fitted functional forms of certain conditional averages of the time series furthermore following a recent work by stolovitzky and ching phys lett a 255 1999 11 we show that the observed pdf can be reproduced by a langevin process with a move dependent noise amplitude the form of the langevin equation can be determined directly from the market data

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

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

measuring volatility with the realized range realized variance being the summation of squared intra day returns has quickly gained popularity as a measure of daily volatility following parkinson 1980 the extreme value method for estimating the variance of the rate of return journal of business 53 61 65 we replace each squared intra day return by the high low range for that period to create a novel and more efficient estimator called the realized range in addition we suggest a bias correction procedure to account for the effects of microstructure frictions based upon scaling the realized range with the average level of the daily range simulation experiments demonstrate that for plausible levels of non trading and bid ask bounce the realized range has a lower mean squared error than the realized variance including variants thereof that are robust to microstructure noise empirical analysis of the s p500 index futures and the s p100 constituents confirms the potential of the realized range

financial options and statistical prediction intervals the paper shows how to convert statistical prediction sets into worst case hedging strategies for derivative securities the prediction sets can in particular be ones for volatilities and correlations of the underlying securities and for interest rates this permits a transfer of statistical conclusions into prices for options and similar financial instruments a prime feature of our results is that one can construct the trading strategy as if the prediction set had a 100 probability if in fact the set has probability 1 the hedging strategy will work with at least the same probability different types of prediction regions are considered the starting value a 0 for the trading strategy corresponding to the 1 prediction region is a form of long term value at risk at the same time a 0 is coherent

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

a non gaussian state space model and application to prediction of records on developpe une classe de modeles qui generalisent le travail de smith 1979 et bathes 1975 pour des donnees censurees

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

local scale models state space alternative to integrated garch processes state space alternative to autoregressive conditional heteroskedasticity models are proposed the initial model which is labelled the gaussian local scale model has a measurement density which is gaussian conditional on the unobservable precision the precision is assumed to be a gamma variable which evolves by being scaled by a beta variable the resulting forecast is a student s t random variable with a scale which is approximately an exponentially weighted moving average ewma of the squares of the past observations the degrees of freedom of the student s t distribution is controlled by the size of the discount parameter of the ewma procedure the gaussianity of the measurement density is potentially inadequate when the model is applied to heavy tailed finance data instead this assumption can be replaced by an exponential power density which allows the observed excess kurtosis to be modelled the choice of the exponential power means that the model still maintains conjugacy so allowing the derivation of an exact filter and likelihood function this model is called the generalised local scale model it has been used to model two exchange rate series

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

bayesian estimation of an extended local scale stochastic volatility model a new version of the local scale model of shephard 1994 is presented its features are identically distributed evolution equation disturbances the incorporation of in the mean effects and the incorporation of variance regressors a bayesian posterior simulator and a new simulation smoother are presented the model is applied to publicly available daily exchange rate and asset return series and is compared with t garch and lognormal stochastic volatility formulations using bayes factors

a fourier transform method for nonparametric estimation of multivariate volatility we provide a nonparametric method for the computation of instantaneous multivariate volatility for continuous semi martingales which is based on fourier analysis the co volatility is reconstructed as a stochastic function of time by establishing a connection between the fourier transform of the prices process and the fourier transform of the co volatility process a non parametric estimator is derived given a discrete unevenly spaced and asynchronously sampled observations of the asset price processes the asymptotic properties of the random estimator are studied namely consistency in probability uniformly in time and convergence in law to a mixture of gaussian distributions

testing forecast optimality under unknown loss empirical tests of forecast optimality have traditionally been conducted under the assumption of mean squared error loss or some other known loss function in this article we establish new testable properties that hold when the forecaster s loss function is unknown but testable restrictions can be imposed on the data generating process trading off conditions on the data generating process against conditions on the loss function we propose flexible estimation of the forecaster s loss function in situations where the loss depends not only on the forecast error but also on other state variables such as the level of the target variable we apply our results to the problem of evaluating the federal reserve s forecasts of output growth forecast optimality is rejected if the fed s loss depends only on the forecast error however the empirical findings are consistent with forecast optimality provided that overpredictions of output growth are costlier to the fed than underpredictions particularly during per