an exact likelihood analysis of the multinomial probit model abstract we develop new methods for conducting a finite sample likelihood based analysis of the multinomial probit model using a variant of the gibbs sampler an algorithm is developed to draw from the exact posterior of the multinomial probit model with correlated errors this approach avoids direct evaluation of the likelihood and thus avoids the problems associated with calculating choice probabilities which affect both the standard likelihood and method of simulated moments approaches both simulated and actual consumer panel data are used to fit six dimensional choice models we also develop methods for analyzing random coefficient and multiperiod probit models

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

limits of conditional expectations convergence of conditional expectations is considered the convergence results for conditional expectations are shown to be the same as when the absolutely continuous change of measure can be made directly a filtering application is examined if the approximating model uses noise increments with a density which must have compact support conditions are given which ensure the expected convergence of the conditional expectations

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

business cycle durations abstract while the development of markov switching extensions to time series modeling has provided a useful way of characterizing business cycle dynamics these models are not without their weaknesses one problem is posed by the fact that since the state space for the unobserved state variables grows with the sample size sampling distributions for maximum likelihood estimates are difficult to establish a second problem is that since the transition probabilities are constant the conditional expected duration of phase is constant this paper extends the model so that the information contained in leading indicator data can be used to forecast transition probabilities these transition probabilities can then be used to calculate expected durations the model is applied to us data to evaluate its ability to explain observed business cycle durations the technical problems encountered with classical techniques are avoided by using bayesian methods gibbs sampling techniques are used to calculate expected posterior durations

a gap result for the norms of semigroups of matrices let be a matrix norm on md c and let a be a finite set of matrices in md c we define mn a to be the maximum norm of a product of n elements of a we show that there is a gap in the possible growth of mn a showing that mn a grows either at least exponentially or is bounded by a polynomial in n of degree at most d 1 moreover we show that the growth is bounded by a polynomial if and only if every element of the semigroup generated by a has all of its eigenvalues on or inside the unit circle

bayes inference in regression models with arma p q errors we develop practical and exact methods of analyzing arma p 4 regression error models in a bayesian framework by using the gibbs sampling and metropolis hastings algorithms and we prove that the kernel of the proposed markov chain sampler converges to the true density the procedures can be applied to pure arma time series models and to determine features of the likelihood function by choosing appropriate diffuse priors our results are unconditional on the initial observations we also show how the algorithm can be further simplified for the important special cases of stationary ar p and invertible ma q models recursive transformations developed in this paper to diagonalize the covariance matrix of the errors should prove useful in frequentist estimation examples with simulated and actual economic data are presented kpj ourls gibbs sampling metropolis hastings algorithm data augmentation time series arma processes markov chain bayesian statistics

convergence of filters with applications to the kalman bucy case for each n and each fixed time t a signal x sup n and a noisy observation y sup n are defined by a pair of stochastic difference equations under certain conditions x sup n y sup n converges in distribution to x y where dx t f t x t dt dv t dy t g t x t dt dw t conditions are found under which convergence in distribution of the conditional expectations e f x sup n mod y sup n to e f x mod y follows for every bounded continuous function f the case in which the conditional expectations still converge but the limit is not e f x mod y is also studied in the situation where f and g are linear functions of x an examination of this limit leads to a kalman bucy type estimate of x sup n which is asymptotically optimal and is an improvement on the usual kalman bucy estimate

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

inference when a nuisance parameter is not identified under the null hypothesis many econometric testing problems involve nuisance parameters which are not identified under the null hypotheses this paper studies the asymptotic distribution theory for such tests the asymptotic distributions of standard test statistics are described as functionals of chi square processes in general the distributions depend upon a large number of unknown parameters we show that a transformation based upon a conditional probability measure yields an asymptotic distribution free of nuisance parameters and we show that this transformation can be easily approximated via simulation the theory is applied to threshold models with special attention given to the so called self exciting threshold autoregressive model monte carlo methods are used to assess the finite sample distributions the tests are applied to u s gnp growth rates and we find that potter s 1995 threshold effect in this series can be possibly explained by sampling variation

the simulation smoother for time series models summary recently suggested procedures for simulating from the posterior density of states given a gaussian state space time series are refined and extended we introduce and study the simulation smoother which draws from the multivariate posterior distribution of the disturbances of the model so avoiding the degeneracies inherent in state samplers the technique is important in gibbs sampling with non gaussian time series models and for performing bayesian analysis of gaussian time series

the boundedness of all products of a pair of matrices is undecidable we show that the boundedness of the set of all products of a given pair sigma of rational matrices is undecidable furthermore we show that the joint or generalized spectral radius rho sigma is not computable because testing whether rho sigma less than or equal to1 is an undecidable problem as a consequence the robust stability of linear systems under time varying perturbations is undecidable and the same is true for the stability of a simple class of hybrid systems we also discuss some connections with the so called finiteness conjecture our results are based on a simple reduction from the emptiness problem for probabilistic finite automata which is known to be undecidable c 2000 elsevier science b v all rights reserved

efficient algorithms for deciding the type of growth of products of integer matrices for a given finite set sigma of matrices with nonnegative integer entries we study the growth with t of max parallel to a 1 a t parallel to a i epsilon sigma we show how to determine in polynomial time whether this growth is bounded polynomial or exponential and we characterize all possible behaviors c 2007 elsevier inc all rights reserved

inference and prediction in a multiple structural break model this paper develops a new bayesian approach to structural break modeling the focuses of the approach are the modeling of in sample structural breaks and forecasting time series allowing out of sample breaks the model has several desirable features first the number of regimes is not fixed but is treated as a random variable second the model adopts a hierarchical prior for regime coefficients which allows for the coefficients of one regime to contain information about coefficients of other regimes third the regime coefficients can be integrated analytically in the posterior density as a consequence the posterior simulator is fast and reliable an application to us real gdp quarterly growth rates links groups of regimes to specific historical periods and provides forecasts of future growth rates

k state switching models with time varying transition distributions does loan growth signal stronger effects of variables on inflation two bayesian sampling schemes are outlined to estimate a time varying markov switching transition distribution using data augmentation transforms the non linear non normal logit transition model into a linear normal one a partial representation of the difference in random utility model in combination with random permutation sampling provides highest sampling efficiency the level of the covariate in the transition distribution which balances the persistence across states is defined to be the threshold level for illustration we estimate a two pillar phillips curve for the euro area in which loan growth affects the transition distribution

sets of matrices all infinite products of which converge an infinite product iit lmi of matrices converges on the right if limi m mi exists a set z ai i l of n x n matrices is called an rcp set rightconvergent product set if all infinite products with each element drawn from z converge such sets of matrices arise in constructing self similar objects like von koch s snowflake curve in various interpolation schemes in constructing wavelets of compact support and in studying nonhomogeneous markov chains this paper gives necessary conditions and also some sufficient conditions for a set x to be an rcp set these are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices necessary and sufficient conditions are given for a finite set z to be an rcp set having a limit function m d rit lad where d d d which is a continuous function on the space of all sequences d with the sequence topology finite rcp sets of column stochastic matrices are completely characterized some results are given on the problem of algorithmically deciding if a given set x is an rcp set

analysis of time series subject to changes in regime abstract this paper introduces an em algorithm for obtaining maximum likelihood estimates of parameters for processes subject to discrete shifts in autoregressive parameters with the shifts themselves modeled as the outcome of a discrete valued markov process the simplicity of the em algorithm permits potential application of the approach to large vector systems

corrigendum addendum to sets of matrices all infinite products of which converge abstract this corrigendum addendum supplies corrected statements and proofs of some results in our paper appearing in linear algebra appl 161 1992 227 263 these results concern special kinds of bounded semigroups of matrices it also reports on progress on the topics of this paper made in the last eight years

bayesian analysis of binary and polychotomous response data abstract a vast literature in statistics biometrics and econometrics is concerned with the analysis of binary and polychotomous response data the classical approach fits a categorical response regression model using maximum likelihood and inferences about the model are based on the associated asymptotic theory the accuracy of classical confidence statements is questionable for small sample sizes in this article exact bayesian methods for modeling categorical response data are developed using the idea of data augmentation the general approach can be summarized as follows the probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data values of the latent data can be simulated from suitable truncated normal distributions if the latent data are known then the posterior distribution of the parameters can be computed using standard results for normal linear models draws from this posterior are used to sample new latent data and t

stochastic linear trends models and estimators abstract the paper considers stochastic linear trends in series with a higher than annual frequency of observation using an approach based on arima models some of the trend models or the model interpretation of trend estimation filters most often found in statistics and econometrics are analysed and compared the properties of the trend optimal estimator are derived and the analysis is extended to seasonally adjusted and or detrended series it is seen that under fairly general conditions the estimator of the unobserved component is noninvertible and will not accept a convergent autoregressive representation this has implications concerning unit root testing and vad model fitting

convergence en loi des suites d integrales stochastiques sur l espace mathbb d 1 de skorokhod for sequences of stochastic integrals k n x n functional limit theorems are presented these results hold under simple natural conditions

estimation and comparison of multiple change point models this paper provides a new bayesian approach for models with multiple change points the centerpiece of the approach is a formulation of the change point model in terms of a latent discrete state variable that indicates the regime from which a particular observation has been drawn this state variable is specified to evolve according to a discrete time discrete state markov process with the transition probabilities constrained so that the state variable can either stay at the current value or jump to the next higher value this parameterization exactly reproduces the change point model the model is estimated by markov chain monte carlo methods using an approach that is based on chib 1996 this methodology is quite valuable since it allows for the fitting of more complex change point models than was possible before methods for the computation of bayes factors are also developed all the techniques are illustrated using simulated and real data sets

dynamic linear models with switching abstract the problem of modeling change in a vector time series is studied using a dynamic linear model with measurement matrices that switch according to a time varying independent random process we derive filtered estimators for the usual state vectors and also for the state occupancy probabilities of the underlying nonstationary measurement process a maximum likelihood estimation procedure is given that uses a pseudo expectation maximization algorithm in the initial stages and nonlinear optimization we relate the models to those considered previously in the literature and give an application involving the tracking of multiple targets

markov chain monte carlo estimation of classical and dynamic switching and mixture models bayesian estimation of a very general model class where the distribution of the observations depends on a latent process taking values in a discrete state space is discussed in this article this model class covers finite mixture modeling markov switching autoregressive modeling and dynamic linear models with switching the consequences the unidentifiability of this type of model has on markov chain monte carlo mcmc estimation are explicitly dealt with joint bayesian estimation of all latent variables model parameters and parameters that determine the probability law of the latent process is carried out by a new mcmc method called permutation sampling the permutation sampler first samples from the unconstrained posterior which often can be done in a convenient multimove manner and then applies a permutation of the current labeling of the states of the latent process in a first run the random permutation sampler used selected the permutation randomly the mcmc output of the random permutation s

bayes regression with autoregressive errors a gibbs sampling approach abstract this paper develops a practical framework for the bayesian analysis of gaussian and student t regression models with autocorrelated errors as is customary in classical estimation procedures the posteriors are conditioned on the initial observations recourse is taken to the method of gibbs sampling an iterative markovian sampling method and it is shown that the proposed approach can readily deal with high order autoregressive processes without requiring an importance sampling function or other tuning constants several examples including one with ar 4 errors are used to illustrate the ideas

a note on marginal stability of switched systems in this note we present criteria for marginal stability and marginal instability of switched systems for switched nonlinear systems we prove that uniform stability is equivalent to the existence of a common weak lyapunov function cwlf that is generally not continuous for switched linear systems we present a unified treatment for marginal stability and marginal instability for both continuous time and discrete time switched systems in particular we prove that any marginally stable system admits a norm as a cwlf by exploiting the largest invariant set contained in a polyhedron several insightful algebraic characteristics are revealed for marginal stability and marginal instability

a note on a gap result for norms of semigroups of matrices abstract let be a matrix norm and be a bounded set of n n complex matrices for each m 1 2 let m sup a a m where m a1a2 am ai bell proved that there is a gap in the possible growth of m if consists of finitely many n n complex matrices in this note based on a lemma by elsner we give an elementary proof of bell s theorem in bounded case

on bayesian analysis of mixtures with an unknown number of components with discussion new methodology for fully bayesian mixture analysis is developed making use of reversible jump markov chain monte carlo methods that are capable of jumping between the parameter subspaces corresponding to different numbers of components in the mixture a sample from the full joint distribution of all unknown variables is thereby generated and this can be used as a basis for a thorough presentation of many aspects of the posterior distribution the methodology is applied here to the analysis of univariate normal mixtures using a hierarchical prior model that offers an approach to dealing with weak prior information while avoiding the mathematical pitfalls of using improper priors in the mixture context

estimation of finite mixture distributions through bayesian sampling summary a formal bayesian analysis of a mixture model usually leads to intractable calculations since the posterior distribution takes into account all the partitions of the sample we present approximation methods which evaluate the posterior distribution and bayes estimators by gibbs sampling relying on the missing data structure of the mixture model the data augmentation method is shown to converge geometrically since a duality principle transfers properties from the discrete missing data chain to the parameters the fully conditional gibbs alternative is shown to be ergodic and geometric convergence is established in the normal case we also consider non informative approximations associated with improper priors assuming that the sample corresponds exactly to a k component mixture

maximum penalized likelihood estimation for independent and markov dependent mixture models summary this paper concerns the use and implementation of maximum penalized likelihood procedures for choosing the number of mixing components and estimating the parameters in independent and markov dependent mixture models computation of the estimates is achieved via algorithms for the automatic generation of starting values for the em algorithm computation of the information matrix is also discussed poisson mixture models are applied to a sequence of counts of movements by a fetal lamb in utero obtained by ultrasound the resulting estimates are seen to provide plausible mechanisms for the physiological process the analysis of count data that are overdispersed relative to the poisson distribution i e variance mean has received considerable recent attention such data might arise in a clinical study in which overdispersion is caused by unexplained or random subject effects alternatively we might observe a time series of counts in which temporal patterns in the data suggest that a poisson model and its implied randomness are inappropriate this paper is motivated by analysis of a time series of overdispersed count data generated in a study of central nervous system development in fetal lambs our data set consists of observed movement counts in 240 consecutive 5 second intervals obtained from a single animal in analysing these data we focus on the use of poisson mixture models assuming independent observations and also markov dependent mixture models or hidden markov models these models assume that the counts follow independent poisson distributions conditional on the rates which are generated from a mixing distribution either independently or with markov dependence we believe finite mixture models are particularly attractive because they provide plausible explanations for variation in the data this paper will emphasize the following issues concerning estimation inference and application of mixture models i choosing the number of model components ii applying the em algorithm to obtain parameter estimates iii generating sufficiently many starting values to identify a global maximum of the likelihood iv avoiding numerical instability

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

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea