Hisashi Tanizaki

Bayesian estimation of state-space models using the Metropolis-Hastings algorithm within Gibbs sampling

In this paper, an attempt is made to show a general solution to nonlinear and/or non-Gaussian state-space modeling in a Bayesian framework, which corresponds to an extension of Carlin et al. (J. Amer. Statist. Assoc. 87(418) (1992) 493-500) and Carter and Kohn (Biometrika 81(3) (1994) 541-553; Biometrika 83(3) (1996) 589-601). Using the Gibbs sampler and the Metropolis-Hastings algorithm, an asymptotically exact estimate of the smoothing mean is obtained from any nonlinear and/or non-Gaussian model. Moreover, taking several candidates of the proposal density function, we examine precision of the proposed Bayes estimator.

Nonlinear and non-Gaussian state-space modeling with Monte Carlo simulations

Abstract We propose two nonlinear and nonnormal filters based on Monte Carlo simulation techniques. In terms of programming and computational requirements both filters are more tractable than other nonlinear filters that use numerical integration, Monte Carlo integration with importance sampling or Gibbs sampling. The proposed filters are extended to prediction and smoothing algorithms. Monte Carlo experiments are carried out to assess the statistical merits of the proposed filters.

Nonlinear and Non-Gaussian State Space Modeling Using Sampling Techniques

In this paper, the nonlinear non-Gaussian filters and smoothers are proposed using the joint density of the state variables, where the sampling techniques such as rejection sampling (RS), importance resampling (IR) and the Metropolis-Hastings independence sampling (MH) are utilized. Utilizing the random draws generated from the joint density, the density-based recursive algorithms on filtering and smoothing can be obtained. Furthermore, taking into account possibility of structural changes and outliers during the estimation period, the appropriately chosen sampling density is possibly introduced into the suggested nonlinear non-Gaussian filtering and smoothing procedures. Finally, through Monte Carlo simulation studies, the suggested filters and smoothers are examined.

On Markov chain Monte Carlo methods for nonlinear and non-Gaussian state-space models

In this paper, a nonlinear and/or non‐Gaussian smoother utilizing Markov chain Monte Carlo Methods is proposed, where the measurement and transition equations are specified in any general formulation and the error terms in the state‐space model are not necessarily normal. The random draws are directly generated from the smoothing densities. For random number generation, the Metropolis‐Hastings algorithm and the Gibbs sampling technique are utilized. The proposed procedure is very simple and easy for programming, compared with the existing nonlinear and non‐Gaussian smoothing techniques. Moreover, taking several candidates of the proposal density function, we examine precision of the proposed estimator.

Bias correction of OLSE in the regression model with lagged dependent variables

It is well known that the ordinary least-squares estimates (OLSE) of autoregressive models are biased in small sample. In this paper, an attempt is made to obtain the unbiased estimates in the sense of median or mean. Using Monte Carlo simulation techniques, we extend the median-unbiased estimator proposed by Andrews (1993, Econometrica 61 (1), 139{165) to the higher-order autoregressive processes, the nonnormal error term and inclusion of any exogenous variables. Also, we introduce the mean-unbiased estimator, which is compared with OLSE and the medium-unbiased estimator. Some simulation studies are performed to examine whether the proposed estimation procedure works well or not, where AR(p) for p=1;2;3 models are examined. We obtain the results that it is possible to recover the true parameter values from OLSE and that the proposed procedure gives us the less-biased estimators than OLSE. Finally, using actually obtained data, an empirical example of the median- and mean-unbiased estimators are shown. c 2000 Elsevier Science B.V. All rights reserved.

On the nonlinear and nonnormal filter using rejection sampling

A nonlinear and/or nonnormal filter is proposed using rejection sampling. Generating random draws of the state-vector directly from the filtering density, the filtering estimate is simply obtained as the arithmetic average of the random draws. In the proposed filter, the random draws are recursively generated at each time. Monte Carlo experiments indicate that the proposed nonlinear and nonnormal filter shows a good performance.

Nonlinear filters based on taylor series expansions

The nonlinear filters based on Taylor series approximation are broadly used for computational simplicity, even though their filtering estimates are clearly biased. In this paper, first, we analyze what is approximated when we apply the expanded nonlinear functions to the standard linear recursive Kalman filter algorithm. Next, since the state variable αt and αt-t are approximated as a conditional normal distribution given information up to time t - 1 (i.e., It-1) in approximation of the Taylor series expansion, it might be appropriate to evaluate each expectation by generating normal random numbers of αt and αt-1 given It-1 and those of the error terms θ and ηt. Thus, we propose the Monte-Carlo simulation filter using normal random draws. Finally we perform two Monte-Carlo experiments, where we obtain the result that the Monte-Carlo simulation filter has a superior performance over the nonlinear filters such as the extended Kalman filter and the second-order nonlinear filter.

Nonlinear and nonnormal filters using Monte Carlo methods

Abstract In this paper, a non-linear and non-normal filter using Monte Carlo simulation techniques is proposed, where the density function derived from the measurement equation and the random draws of the state-vector generated from the transition equation are utilized. The proposed filter has less computational burden and easier programming than the other non-linear and non-normal filters such as the numerical integration procedure and the Monte Carlo integration approach. Furthermore, the proposed filter is extended to prediction and smoothing algorithms. Finally, by Monte Carlo experiments, we compare the non-linear and non-normal procedures.

Asymptotically exact confidence intervals of CUSUM and CUSUMSQ tests : a numerical derivation using simulation technique

In testing a structural change, the approximated confidence intervals are conventionally used for CUSUM and CUSUMSQ tests. This paper numerically derives the asymptotically exact confidence intervals of CUSUM and CUSUMSQ tests. It can be easily extended to nonnormal and or nonlinear models

Note on the Sampling Distribution for the Metropolis-Hastings Algorithm

Abstract The Metropolis-Hastings algorithm has been important in the recent development of Bayes methods. This algorithm generates random draws from a target distribution utilizing a sampling (or proposal) distribution. This article compares the properties of three sampling distributions—the independence chain, the random walk chain, and the Taylored chain suggested by Geweke and Tanizaki (Geweke, J., Tanizaki, H. (1999). On Markov Chain Monte-Carlo methods for nonlinear and non-Gaussian state-space models. Communications in Statistics, Simulation and Computation 28(4):867–894, Geweke, J., Tanizaki, H. (2001). Bayesian estimation of state-space model using the Metropolis-Hastings algorithm within Gibbs sampling. Computational Statistics and Data Analysis 37(2):151–170).