In this study, we propose a nonlinear Bayesian extension of the Lee–Carter (LC) model using a single-stage procedure with a dimensionality reduction neural network (NN). LC is originally estimated using a two-stage procedure: dimensionality reduction of data by singular value decomposition followed by a time series model fitting. To address the limitations of LC, which are attributed to the two-stage estimation and insufficient model fitness to data, single-stage procedures using the Bayesian state-space (BSS) approaches and extensions of flexibility in modeling by NNs have been proposed. As a fusion of these two approaches, we propose a NN extension of LC with a variational autoencoder that performs the variational Bayesian estimation of a state-space model and dimensionality reduction by autoencoding. Despite being a NN model that performs single-stage estimation of parameters, our model has excellent interpretability and the ability to forecast with confidence intervals, as with the BSS models, without using Markov chain Monte Carlo methods.

A new class of time series models is used to track the progress of the COVID-19 epidemic in the UK in early 2021. Models are fitted to England and the regions, as well as to the UK as a whole. The growth rate of the daily number of cases and the instantaneous reproduction number are computed regularly and compared with those produced by SAGE. The results from figures published each day are compared with results based on figures by specimen date, which may be more accurate but are subject to substantial revisions. It is then shown how data from the two different sources can be combined in bivariate models.

Cohort effects are important factors in determining the evolution of human mortality for certain countries. Extensions of dynamic mortality models with cohort features have been proposed in the literature to account for these factors under the generalised linear modelling framework. In this paper we approach the problem of mortality modelling with cohort factors incorporated through a novel formulation under a state-space methodology. In the process we demonstrate that cohort factors can be formulated naturally under the state-space framework, despite the fact that cohort factors are indexed according to year-of-birth rather than year. Bayesian inference for cohort models in a state-space formulation is then developed based on an efficient Markov chain Monte Carlo sampler, allowing for the quantification of parameter uncertainty in cohort models and resulting mortality forecasts that are used for life expectancy and life table constructions. The effectiveness of our approach is examined through comprehensive empirical studies involving male and female populations from various countries. Our results show that cohort patterns are present for certain countries that we studied and the inclusion of cohort factors are crucial in capturing these phenomena, thus highlighting the benefits of introducing cohort models in the state-space framework. Forecasting of cohort models is also discussed in light of the projection of cohort factors.

A prevalent problem in general state-space models is the approximation of the smoothing distribution of a state conditional on the observations from the past, the present, and the future. The aim of this paper is to provide a rigorous analysis of such approximations of smoothed distributions provided by the two-filter algorithms. We extend the results available for the approximation of smoothing distributions to these two-filter approaches which combine a forward filter approximating the filtering distributions with a backward information filter approximating a quantity proportional to the posterior distribution of the state, given future observations.

We analyse the performance of a recursive Monte Carlo method for the Bayesian estimation of the static parameters of a discrete-time state-space Markov model. The algorithm employs two layers of particle filters to approximate the posterior probability distribution of the model parameters. In particular, the first layer yields an empirical distribution of samples on the parameter space, while the filters in the second layer are auxiliary devices to approximate the (analytically intractable) likelihood of the parameters. This approach relates the novel algorithm to the recent sequential Monte Carlo square method, which provides a nonrecursive solution to the same problem. In this paper we investigate the approximation of integrals of real bounded functions with respect to the posterior distribution of the system parameters. Under assumptions related to the compactness of the parameter support and the stability and continuity of the sequence of posterior distributions for the state-space model, we prove that the Lp norms of the approximation errors vanish asymptotically (as the number of Monte Carlo samples generated by the algorithm increases) and uniformly over time. We also prove that, under the same assumptions, the proposed scheme can asymptotically identify the parameter values for a class of models. We conclude the paper with a numerical example that illustrates the uniform convergence results by exploring the accuracy and stability of the proposed algorithm operating with long sequences of observations.

This paper explores and develops alternative statistical representations and estimation approaches for dynamic mortality models. The framework we adopt is to reinterpret popular mortality models such as the Lee–Carter class of models in a general state-space modelling methodology, which allows modelling, estimation and forecasting of mortality under a unified framework. We propose alternative model identification constraints which are more suited to statistical inference in filtering and parameter estimation. We then develop a class of Bayesian state-space models which incorporate a priori beliefs about the mortality model characteristics as well as for more flexible and appropriate assumptions relating to heteroscedasticity that present in observed mortality data. To study long-term mortality dynamics, we introduce stochastic volatility to the period effect. The estimation of the resulting stochastic volatility model of mortality is performed using a recent class of Monte Carlo procedure known as the class of particle Markov chain Monte Carlo methods. We illustrate the framework using Danish male mortality data, and show that incorporating heteroscedasticity and stochastic volatility markedly improves model fit despite an increase of model complexity. Forecasting properties of the enhanced models are examined with long-term and short-term calibration periods on the reconstruction of life tables.

We consider the numerical approximation of the filtering problem in high dimensions, that is, when the hidden state lies in ℝd with large d. For low-dimensional problems, one of the most popular numerical procedures for consistent inference is the class of approximations termed particle filters or sequential Monte Carlo methods. However, in high dimensions, standard particle filters (e.g. the bootstrap particle filter) can have a cost that is exponential in d for the algorithm to be stable in an appropriate sense. We develop a new particle filter, called the space‒time particle filter, for a specific family of state-space models in discrete time. This new class of particle filters provides consistent Monte Carlo estimates for any fixed d, as do standard particle filters. Moreover, when there is a spatial mixing element in the dimension of the state vector, the space‒time particle filter will scale much better with d than the standard filter for a class of filtering problems. We illustrate this analytically for a model of a simple independent and identically distributed structure and a model of an L-Markovian structure (L≥ 1, L independent of d) in the d-dimensional space direction, when we show that the algorithm exhibits certain stability properties as d increases at a cost 𝒪(nNd2), where n is the time parameter and N is the number of Monte Carlo samples, which are fixed and independent of d. Our theoretical results are also supported by numerical simulations on practical models of complex structures. The results suggest that it is indeed possible to tackle some high-dimensional filtering problems using the space‒time particle filter that standard particle filters cannot handle.

This paper employs the state-space model to reexamine the fundamental issue in finance of whether it is the expected returns or the expected dividends growth that is primarily responsible for stock price variations. We use Bayesian methods to show that there is a substantial uncertainty about the contributions of expected returns and expected dividends to fluctuations in the price–dividend ratio when the aggregate returns and dividends data are used. The substantial uncertainty of the contributions results from the model being weakly identified. Our finding challenges the notion long held in the existing literature that it is the expected returns that contribute most to price–dividend variations.

A statistical approach to evaluate the temporal trends in the abundance of female Centroscymnus coelolepis in Portuguese waters of ICES division IXa is presented. A state space model is used, which integrates all the available information of the species’ life history as well as knowledge of its biological dynamics. The model involves two processes that run in parallel: a non-observed process (the state process) that describes the annual female population abundance and an observational process of annual fisheries catches in numbers, assumed to be measured with error. Estimation is done within the Bayesian paradigm using sequential importance sampling with resampling. To evaluate the sensitivity of the model to the prior distributions chosen for the parameters, three scenarios with different levels of prior information were considered. Trends in population abundance level and the abundance levels themselves are quite similar in the two scenarios using biological information, but the model that incorporated all the available biological information in the priors provided the best fit to the observed data. The results indicate that taking into account the main biological drivers and the fishing information in the same state space model provides a coherent picture of the population abundance trends, further suggesting that the fishing impact on the population inhabiting Portuguese mainland waters was low.

We first explore empirical evidence of parameter and shock uncertainties in a state-space model with Markov switching. The evidence indicates that uncertainties in the U.S. economy have been too great to accurately define monetary policy rules. We then explore monetary policy rules under uncertainty with two approaches: the RLS learning algorithm and robust control. The former allows the parameters to be learned for a given model. Yet, as our results of the RLS learning in a framework of optimal control indicate, the state variables do not necessarily converge even in a nonstochastic model. The latter, by permitting uncertainty with respect to model misspecification, allows for a broader framework. Our study on robust control shows that robust optimal monetary policy rules reveal a stronger response to fluctuations in inflation and output than when no uncertainty exists, implying that uncertainty does not necessarily require caution.