Simulation Smoother Research Articles

Comparison of simulation-based algorithms for parameter estimation and state reconstruction in nonlinear state-space models

<p style='text-indent:20px;'>This study aims at comparing simulation-based approaches for estimating both the state and unknown parameters in nonlinear state-space models. Numerical results on different toy models show that the combination of a Conditional Particle Filter (CPF) with Backward Simulation (BS) smoother and a Stochastic Expectation-Maximization (SEM) algorithm is a promising approach. The CPFBS smoother run with a small number of particles allows to explore efficiently the state-space and simulate relevant trajectories of the state conditionally to the observations. When combined with the SEM algorithm, this algorithm provides accurate estimates of the state and the parameters in nonlinear models, where the application of EM algorithms combined with a standard particle smoother or an ensemble Kalman smoother is limited.</p>

A Fast and Efficient Markov Chain Monte Carlo Method for Market Microstructure Model

The non-linear market microstructure (MM) model for financial time series modeling is a flexible stochastic volatility model with demand surplus and market liquidity. The estimation of the model is difficult, since the unobservable surplus demand is a time-varying stochastic variable in the return equation, and the market liquidity arises both in the mean term and in the variance term of the return equation in the MM model. A fast and efficient Markov Chain Monte Carlo (MCMC) approach based on an efficient simulation smoother algorithm and an acceptance-rejection Metropolis–Hastings algorithm is designed to estimate the non-linear MM model. Since the simulation smoother algorithm makes use of the band diagonal structure and positive definition of Hessian matrix of the logarithmic density, it can quickly draw the market liquidity. In addition, we discuss the MM model with Student-t heavy tail distribution that can be utilized to address the presence of outliers in typical financial time series. Using the presented modeling method to make analysis of daily income of the S&P 500 index through the point forecast and the density forecast, we find clear support for time-varying volatility, volatility feedback effect, market microstructure theory, and Student-t heavy tails in the financial time series. Through this method, one can use the estimated market liquidity and surplus demand which is much smoother than the strong stochastic return process to assist the transaction decision making in the financial market.

Estimation of time-varying autoregressive stochastic volatility models with stable innovations

A new time-varying autoregressive stochastic volatility model with $$\alpha $$ -stable innovations (TVAR $$\alpha $$ SV) is proposed. This new model for time series data combines a time-varying autoregressive component and a stochastic scaling as known from stochastic volatility models with $$\alpha $$ -stable distributed noise. Hence, the model can cover extreme events better than classical stochastic volatility models. Furthermore, we develop a Gibbs sampling procedure for the estimation of the model parameters. The procedure is based on the estimation strategy by Kim et al. (Rev Econ Stud 65(3): 361–393, 1998) for classical stochastic volatility models, however, the estimation procedure requires a deliberate approximation of $$\alpha $$ -stable distributions by finite mixtures of normal distributions and the application of a simulation smoother for linear Gaussian state space models. A simulation study for the new estimation procedure illustrates the appealing accuracy. Finally, we apply the model to electricity spot price data.

Testing for non-chaoticity under noisy dynamics using the largest Lyapunov exponent

In this paper, we introduce a robust procedure to test for non-chaoticity when data are contaminated by an additive noise. Under the Kalman filter framework, our procedure first amounts to compute the largest Lyapunov exponent of the extracted signal. The exponent describes the log-divergence of a dynamical system (Rosenstein et al. in Phys D 65(1–2):117–134, 1993. https://doi.org/10.1016/0167-2789(93)90009-p). Then, using the so-called simulation smoother, we generate a high number of trajectories of the state-vector, conditional on the observed series, and compute the empirical distribution of the largest Lyapunov exponent. The distribution allows for computing confidence intervals. We can thus test if the largest Lyapunov exponent is not significantly greater than zero. Using Monte Carlo simulations, we show the validity of such an approach. We provide an illustration using toy models, which depict several dynamical systems. Finally, we implement tests of non-chaoticity on financial time series. We find no empirical evidence of chaotic patterns. Our approach is simple, efficient, and tests for chaos when data are measured with errors (i.e., noisy dynamics).

A simulation smoother for long memory time series with correlated and heteroskedastic additive noise

The purpose of this paper is to extend the work of So (1999) by accommodating heteroskedasticity in long memory processes and correlation in disturbances. We propose an alternative representation of long memory processes from which we develop filtering equations, prediction densities and a simulation smoother for latent state variables as in the classical Kalman filter. We illustrate the simulation smoother introduced in this paper by estimating a class of long memory models with heteroskedastic correlated additive noises.

A Flexible State-Space Model with Lagged States and Lagged Dependent Variables: Simulation Smoothing

We provide a simulation smoother to a exible state-space model with lagged states and lagged dependent variables. Qian (2014) has introduced this state-space model and proposes a fast Kalman filter with time-varying state dimension in the presence of missing observations in the data. In this paper, we derive the corresponding Kalman smoother moments and propose an efficient simulation smoother, which relies on mean corrections for unconditional vectors. When applied to a factor model, the proposed simulation smoother for the states is efficient compared to other state-space models without lagged states and/or lagged dependent variables in terms of computing time.

On Importance Sampling for State Space Models

We consider likelihood inference and state estimation by means of importance sampling for state space models with a nonlinear non-Gaussian observation y ~ p(y|alpha) and a linear Gaussian state alpha ~ p(alpha). The importance density is chosen to be the Laplace approximation of the smoothing density p(alpha|y). We show that computationally efficient state space methods can be used to perform all necessary computations in all situations. It requires new derivations of the Kalman filter and smoother and the simulation smoother which do not rely on a linear Gaussian observation equation. Furthermore, results are presented that lead to a more effective implementation of importance sampling for state space models. An illustration is given for the stochastic volatility model with leverage.

A computationally efficient method for vector autoregression with mixed frequency data

A linear transformation method is proposed to handle the vector autoregression with mixed frequency time series data. Temporally aggregated observations impose linear constraints on the distribution of latent variables, which are converted such that each observation replaces a latent variable. Full-sample transformation yields a closed-form simulation smoother, while partial-sample transformation leads to a computationally efficient sampler suitable for parallel computing.

A note on implementing the Durbin and Koopman simulation smoother

The correct implementation of the Durbin and Koopman simulation smoother is explained. A possible misunderstanding is pointed out and clarified for both the basic state space model with a non-zero mean of the initial state and with time-varying intercepts (mean adjustments).

Estimating a DSGE model for Japan in a data-rich environment

A dynamic factor model (DFM), widely used in empirical research in macroeconomics, shows that common factors extracted from large panel data sets are key factors behind the fluctuations of primal macroeconomic series. Boivin and Giannoni (2006) and Kryshko (2011) combine a dynamic stochastic general equilibrium (DSGE) model with a DFM as a data-rich DSGE model, in which model variables are regarded as common factors derived from large data sets. Following Smets and Wouters (2003, 2007), we estimate a new Keynesian DSGE model for Japan between 1981Q1 and 1995Q4 in a data-rich environment with 55 macroeconomic indicators using Markov chain Monte Carlo (MCMC) methods. Using a simulation smoother developed by de Jong and Shephard (1995), unlike previous studies, we succeeded in sampling model variables and exogenous shocks used for analyzing sources of business cycles. We found that a data-rich DSGE model with an inappropriate data set or inaccurate specificities reduces efficiency even though the number of indicators is fulfilling.

Nonlinear State Space Smoothing Using the Conditional Particle Filter

To estimate the smoothing distribution in a nonlinear state space model, we apply the conditional particle filter with ancestor sampling. This gives an iterative algorithm in a Markov chain Monte Carlo fashion, with asymptotic convergence results. The computational complexity is analyzed, and our proposed algorithm is successfully applied to the challenging problem of sensor fusion between ultrawideband and accelerometer/gyroscope measurements for indoor positioning. It appears to be a competitive alternative to existing nonlinear smoothing algorithms, in particular the forward filtering-backward simulation smoother.

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.

Stochastic functional data analysis: a diffusion model-based approach.

This article presents a new modeling strategy in functional data analysis. We consider the problem of estimating an unknown smooth function given functional data with noise. The unknown function is treated as the realization of a stochastic process, which is incorporated into a diffusion model. The method of smoothing spline estimation is connected to a special case of this approach. The resulting models offer great flexibility to capture the dynamic features of functional data, and allow straightforward and meaningful interpretation. The likelihood of the models is derived with Euler approximation and data augmentation. A unified Bayesian inference method is carried out via a Markov chain Monte Carlo algorithm including a simulation smoother. The proposed models and methods are illustrated on some prostate-specific antigen data, where we also show how the models can be used for forecasting.

Efficient Bayesian estimation of multivariate state space models

A Bayesian Markov chain Monte Carlo methodology is developed for the estimation of multivariate linear Gaussian state space models. In particular, an efficient simulation smoothing algorithm is proposed that makes use of the univariate representation of the state space model. Substantial gains over existing algorithms in computational efficiency are achieved using the new simulation smoother for the analysis of high dimensional multivariate time series. The methodology is used to analyse a multivariate time series dataset of the Normalised Difference Vegetation Index (NDVI), which is a proxy for the level of live vegetation, for a particular grazing property located in Queensland, Australia.

Transformations and seasonal adjustment

Abstract. We address the problem of seasonal adjustment of a nonlinear transformation of the original time series, measured on a ratio scale, which aims at enforcing two essential features: additivity and orthogonality of the components. The posterior mean and variance of the seasonally adjusted series admit an analytic finite representation only for particular values of the transformation parameter, e.g. for a fractional Box–Cox transformation parameter. Even if available, the analytical derivation can be tedious and difficult. As an alternative we propose to compute the two conditional moments of the seasonally adjusted series by means of numerical and Monte Carlo integration. The former is both fast and reliable in univariate applications. The latter uses the algorithm known as the ‘simulation smoother’ and it is most useful in multivariate applications. We present two case studies dealing with robust seasonal adjustment under the square root and the fourth root transformation. Our overall conclusion is that robust seasonal adjustment under transformations is feasible from the computational standpoint and that the possibility of transforming the scale ought to be considered as a further option for improving the quality of seasonal adjustment.

Block sampler and posterior mode estimation for asymmetric stochastic volatility models

A new efficient simulation smoother and disturbance smoother are introduced for asymmetric stochastic volatility models where there exists a correlation between today's return and tomorrow's volatility. The state vector is divided into several blocks where each block consists of many state variables. For each block, corresponding disturbances are sampled simultaneously from their conditional posterior distribution. The algorithm is based on the multivariate normal approximation of the conditional posterior density and exploits a conventional simulation smoother for a linear and Gaussian state-space model. The performance of our method is illustrated using two examples: (1) simple asymmetric stochastic volatility model and (2) asymmetric stochastic volatility model with state-dependent variances. The popular single move sampler which samples a state variable at a time is also conducted for comparison in the first example. It is shown that our proposed sampler produces considerable improvement in the mixing property of the Markov chain Monte Carlo chain.

State space mixed models for longitudinal observations with binary and binomial responses

We propose a new class of state space models for longitudinal discrete response data where the observation equation is specified in an additive form involving both deterministic and random linear predictors. These models allow us to explicitly address the effects of trend, seasonal or other time-varying covariates while preserving the power of state space models in modeling serial dependence in the data. We develop a Markov chain Monte Carlo algorithm to carry out statistical inference for models with binary and binomial responses, in which we invoke de Jong and Shephard’s (Biometrika 82(2):339–350, 1995) simulation smoother to establish an efficient sampling procedure for the state variables. To quantify and control the sensitivity of posteriors on the priors of variance parameters, we add a signal-to-noise ratio type parameter in the specification of these priors. Finally, we illustrate the applicability of the proposed state space mixed models for longitudinal binomial response data in both simulation studies and data examples.

Classical and Bayesian Analysis of Univariate and Multivariate Stochastic Volatility Models

In this paper, efficient importance sampling (EIS) is used to perform a classical and Bayesian analysis of univariate and multivariate stochastic volatility (SV) models for financial return series. EIS provides a highly generic and very accurate procedure for the Monte Carlo (MC) evaluation of high-dimensional interdependent integrals. It can be used to carry out ML-estimation of SV models as well as simulation smoothing where the latent volatilities are sampled at once. Based on this EIS simulation smoother, a Bayesian Markov chain Monte Carlo (MCMC) posterior analysis of the parameters of SV models can be performed.

A simple and efficient simulation smoother for state space time series analysis

A simulation smoother in state space time series analysis is a procedure for drawing samples from the conditional distribution of state or disturbance vectors given the observations. We present a new technique for this which is both simple and computationally efficient. The treatment includes models with diffuse initial conditions and regression effects. Computational comparisons are made with the previous standard method. Two applications are provided to illustrate the use of the simulation smoother for Gibbs sampling for Bayesian inference and importance sampling for classical inference. © 2002 Biometrika Trust.

Bayesian Inference of Long-Memory Stochastic Volatility via Wavelets

In this paper we are concerned with estimating the fractional order of integration associated with a long-memory stochastic volatility model. We develop a new Bayesian estimator based on the Markov chain Monte Carlo sampler and the wavelet representation of the log-squared returns to draw values of the fractional order of integration and latent volatilities from their joint posterior distribution. Unlike short-memory stochastic volatility models, long-memory stochastic volatility models do not have a state-space representation, and thus their sampler cannot employ the Kalman filters simulation smoother to update the chain of latent volatilities. Instead, we design a simulator where the latent long-memory volatilities are drawn quickly and efficiently from the near independent multivariate distribution of the long-memory volatility's wavelet coefficients. We find that sampling volatility in the wavelet domain, rather than in the time domain, leads to a fast and simulation-efficient sampler of the posterior distribution for the volatility's long-memory parameter and serves as a promising alternative estimator to the existing frequentist based estimators of long-memory volatility.