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

marginal likelihood from the metropolis hastings output this article provides a framework for estimating the marginal likelihood for the purpose of bayesian model comparisons the approach extends and completes the method presented in chib 1995 by overcoming the problems associated with the presence of intractable full conditional densities the proposed method is developed in the context of mcmc chains produced by the metropolis hastings algorithm whose building blocks are used both for sampling and marginal likelihood estimation thus economizing on prerun tuning effort and programming experiments involving the logit model for binary data hierarchical random effects model for clustered gaussian data poisson regression model for clustered count data and the multivariate probit model for correlated binary data are used to illustrate the performance and implementation of the method these examples demonstrate that the method is practical and widely applicable

high resolution quantization theory and the vector quantizer advantage the authors consider how much performance advantage a fixed dimensional vector quantizer can gain over a scalar quantizer they collect several results from high resolution or asymptotic in rate quantization theory and use them to identify source and system characteristics that contribute to the vector quantizer advantage one well known advantage is due to improvement in the space filling properties of polytopes as the dimension increases others depend on the source s memory and marginal density shape the advantages are used to gain insight into product transform lattice predictive pyramid and universal quantizers although numerical prediction consistently overestimated gains in low rate 1 bit sample experiments the theoretical insights may be useful even at these rates

getting it right joint distribution tests of posterior simulators analytical or coding errors in posterior simulators can produce reasonable but incorrect approximations of posterior moments this article develops simple tests of posterior simulators that detect both kinds of errors and uses them to detect and correct errors in two previously published articles the tests exploit the fact that a bayesian model specifies the joint distribution of observables data and unobservables parameters there are two joint distribution simulators the marginal conditional simulator draws unobservables from the prior and then observables conditional on unobservables the successive conditional simulator alternates between the posterior simulator and an observables simulator formal comparison of moment approximations of the two simulators reveals existing analytical or coding errors in the posterior simulator

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

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

computing bayes factors by combining simulation and asymptotic approximations abstract the bayes factor is a ratio of two posterior normalizing constants which may be difficult to compute we compare several methods of estimating bayes factors when it is possible to simulate observations from the posterior distributions via markov chain monte carlo or other techniques the methods that we study are all easily applied without consideration of special features of the problem provided that each posterior distribution is well behaved in the sense of having a single dominant mode we consider a simulated version of laplace s method a simulated version of bartlett correction importance sampling and a reciprocal importance sampling technique we also introduce local volume corrections for each of these in addition we apply the bridge sampling method of meng and wong we find that a simulated version of laplace s method with local volume correction furnishes an accurate approximation that is especially useful when likelihood function evaluations are costly a simple bridge sampli

trends and cycles in economic time series a bayesian approach trends and cyclical components in economic time series are modeled in a bayesian framework this enables prior notions about the duration of cycles to be used while the generalized class of stochastic cycles employed allows the possibility of relatively smooth cycles being extracted the posterior distributions of such underlying cycles can be very informative for policy makers particularly with regard to the size and direction of the output gap and potential turning points from the technical point of view a contribution is made in investigating the most appropriate prior distributions for the parameters in the cyclical components and in developing markov chain monte carlo methods for both univariate and multivariate models applications to us macroeconomic series are presented

analysis of multivariate probit models summary this paper provides a practical simulation based bayesian and non bayesian analysis of correlated binary data using the multivariate probit model the posterior distribution is simulated by markov chain monte carlo methods and maximum likelihood estimates are obtained by a monte carlo version of the em algorithm a practical approach for the computation of bayes factors from the simulation output is also developed the methods are applied to a dataset with a bivariate binary response to a four year longitudinal dataset from the six cities study of the health effects of air pollution and to a sevenvariate binary response dataset on the labour supply of married women from the panel survey of income dynamics

stochastic volatility with leverage fast and efficient likelihood inference this paper is concerned with the bayesian analysis of stochastic volatility sv models with leverage specifically the paper shows how the often used kim et al 1998 stochastic volatility likelihood inference and comparison with arch models review of economic studies 65 361 393 method that was developed for sv models without leverage can be extended to models with leverage the approach relies on the novel idea of approximating the joint distribution of the outcome and volatility innovations by a suitably constructed ten component mixture of bivariate normal distributions the resulting posterior distribution is summarized by mcmc methods and the small approximation error in working with the mixture approximation is corrected by a reweighting procedure the overall procedure is fast and highly efficient we illustrate the ideas on daily returns of the tokyo stock price index finally extensions of the method are described for superposition models where the log volatility is made up of a linear combination of heterogenous and independent autoregressions and heavy tailed error distributions student and log normal

markov chain monte carlo methods for stochastic volatility models this paper is concerned with simulation based inference in generalized models of stochastic volatility defined by heavy tailed student t distributions with unknown degrees of freedom and exogenous variables in the observation and volatility equations and a jump component in the observation equation by building on the work of kim shephard and chib rev econom stud 65 1998 361 we develop efficient markov chain monte carlo algorithms for estimating these models the paper also discusses how the likelihood function of these models can be computed by appropriate particle filter methods computation of the marginal likelihood by the method of chib j amer statist assoc 90 1995 1313 is also considered the methodology is extensively tested and validated on simulated data and then applied in detail to daily returns data on the s p 500 index where several stochastic volatility models are formally compared under different priors on the parameters

nonlinear state space models with state dependent variances nonlinear state space models with state dependent variances sdvs are commonly used in financial time series important examples include stochastic volatility sv and affine term structure models we propose a methodology for state smoothing in this class of models our smoothing technique is simulation based and uses an auxiliary mixture model key features of the auxiliary mixture model are the use of state dependent weights and efficient block sampling algorithms to jointly update all unobserved states given latent mixture indicators conditional on latent indicator variables the auxiliary mixture model reduces to a normal dynamic linear model we illustrate our methodology with two time series applications first we show how to construct the auxiliary model for a logarithmic sv model and compare the performance of our methodology with the current literature next we implement a square root sv model with jumps for short term interest rates in hong kong

bennett s integral for vector quantizers this paper extends bennett s 1948 integral from scalar to vector quantizers giving a simple formula that expresses the rth power distortion of a many point vector quantizer in terms of the number of points point density function inertial profile and the distribution of the source the inertial profile specifies the normalized moment of inertia of quantization cells as a function of location the extension is formulated in terms of a sequence of quantizers whose point density and inertial profile approach known functions as the number of points increase precise conditions are given for the convergence of distortion suitably normalized to bennett s integral previous extensions did not include the inertial profile and consequently provided only bounds or applied only to quantizers with congruent cells such as lattice and optimal quantizers the new version of bennett s integral provides a framework for the analysis of suboptimal structured vector quantizers it is shown how the loss in performance of such quantizers relative to optimal unstructured ones can be decomposed into point density and cell shape losses as examples these losses are computed for product quantizers and used to gain further understanding of the performance of scalar quantizers applied to stationary memoryless sources and of transform codes applied to gaussian sources with memory it is shown that the short coming of such quantizers is that they must compromise between point density and cell shapes

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

two results on the asymptotic performance of quantizers a necessary and sufficient condition is presented for the normalized asymptotic r th power distortion of a mismatched multidimensional quantizer to converge to a certain optimum constant known as bennett s integral the dimensionality of our results is allowed to approach infinity in order to make some universal source coding comparisons to quantization theory

marginal likelihood from the gibbs output abstract in the context of bayes estimation via gibbs sampling with or without data augmentation a simple approach is developed for computing the marginal density of the sample data marginal likelihood given parameter draws from the posterior distribution consequently bayes factors for model comparisons can be routinely computed as a by product of the simulation hitherto this calculation has proved extremely challenging our approach exploits the fact that the marginal density can be expressed as the prior times the likelihood function over the posterior density this simple identity holds for any parameter value an estimate of the posterior density is shown to be available if all complete conditional densities used in the gibbs sampler have closed form expressions to improve accuracy the posterior density is estimated at a high density point and the numerical standard error of resulting estimate is derived the ideas are applied to probit regression and finite mixture models

a simple class of asymptotically optimal quantizers a simple class of quantizers is introduced which are asymptotically optimal as the number of quantization levels increases to infinity with respect to a mean r th power absolute error distortion measure these asymptotically optimal quantizers are very easy to compute their performance is evaluated for several distributions and compares favorably with the performance of the optimal quantizers in all cases for which the latter have been computed in addition their asymptotic robustness is studied under location scale and shape mismatch for several families of distributions

existence of positive ground state solutions for kirchhoff type problems abstract in this paper we study the existence of positive ground state solutions for the nonlinear kirchhoff type problem a b r 3 u 2 u v x u f u in r 3 u h 1 r 3 u 0 in r 3 where a b 0 are constants f c r r is subcritical near infinity and superlinear near zero and satisfies the berestycki lions condition by using an abstract critical point theorem established by jeanjean and a new global compactness lemma we show that the above problem has at least a positive ground state solution our result generalizes the results of li and ye 2014 concerning the nonlinearity f u u p 1 u with p 2 5

on leverage in a stochastic volatility model this paper is concerned with the specification for modelling financial leverage effect in the context of stochastic volatility sv models two alternative specifications co exist in the literature one is the euler approximation to the well known continuous time sv model with leverage effect and the other is the discrete time sv model of jacquier et al j econometrics 122 2004 185 using a gaussian nonlinear state space form with uncorrelated measurement and transition errors i show that it is easy to interpret the leverage effect in the conventional model whereas it is not clear how to obtain and interpret the leverage effect in the model of jacquier et al empirical comparisons of these two models via bayesian markov chain monte carlo mcmc methods further reveal that the specification of jacquier et al is inferior simulation experiments are conducted to study the sampling properties of bayes mcmc for the conventional model

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

bayesian model choice asymptotics and exact calculations abstract model determination is a fundamental data analytic task here we consider the problem of choosing amongst a finite with loss of generality we assume two set of models after briefly reviewing classical and bayesian model choice strategies we present a general predictive density which includes all proposed bayesian approaches we are aware of using laplace approximations we can conveniently assess and compare asymptotic behavior of these approaches concern regarding the accuracy of these approximation for small to moderate sample sizes encourages the use of monte carlo techniques to carry out exact calculations a data set fit with nested non linear models enables comparison between proposals and between exact and asymptotic values

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

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

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 result on a class of elliptic equations involving kirchhoff type nonlocal term abstract in the paper we consider the following elliptic equations involving kirchhoff type nonlocal term 0 1 a b r 3 u 2 d x u v x u f u x r 3 in which a 0 b 0 the nonlinear term f satisfies the berestycki lions conditions under some assumptions on v we obtained equation eq 0 1 has at least a solution

calculating posterior distributions and modal estimates in markov mixture models abstract this paper is concerned with finite mixture models in which the populations from one observation to the next are selected according to an unobserved markov process a new full bayesian approach based on the method of gibbs sampling is developed calculations are simplified by data augmentation achieved by introducing a population index variable into the list of unknown parameters it is shown that the latent variables one for each observation can be simulated from their joint distribution given the data and the remaining parameters this result serves to accelerate the convergence of the gibbs sample modal estimates are also computed by stochastic versions of the em algorithm these provide an alternative to a full bayesian approach and to existing methods of locating the maximum likelihood estimate the ideas are applied in detail to poisson data mixtures of multivariate normal distributions and autoregressive time series

bayesian analysis of stochastic volatility models with fat tails and correlated errors abstract the basic univariate stochastic volatility model specifies that conditional volatility follows a log normal auto regressive model with innovations assumed to be independent of the innovations in the conditional mean equation since the introduction of practical methods for inference in the basic volatility model j business econom statist 12 4 371 it has been observed that the basic model is too restrictive for many financial series we extend the basic svol to allow for a so called leverage effect via correlation between the volatility and mean innovations and for fat tails in the mean equation innovation a bayesian markov chain monte carlo algorithm is developed for the extended volatility model thus far likelihood based inference for the correlated svol model has not appeared in the literature we develop bayes factors to assess the importance of the leverage and fat tail extensions sampling experiments reveal little loss in precision from adding the model extensions but a large loss from using the basic model in the presence of mis specification there is overwhelming evidence of a leverage effect for weekly and daily equity indices the evidence in favor of fat tails is very strong for daily exchange rate and equity indices but less so for weekly data we also find that volatility estimates from the extended model are markedly different from those produced by the basic svol