A New Approach to Volatility Modeling: The Factorial Hidden Markov Volatility Model

A new process—the factorial hidden Markov volatility (FHMV) model—is proposed to model financial returns or realized variances. Its dynamics are driven by a latent volatility process specified as a product of three components: a Markov chain controlling volatility persistence, an independent discrete process capable of generating jumps in the volatility, and a predictable (data-driven) process capturing the leverage effect. An economic interpretation is attached to each one of these components. Moreover, the Markov chain and jump components allow volatility to switch abruptly between thousands of states, and the transition matrix of the model is structured to generate a high degree of volatility persistence. An empirical study on six financial time series shows that the FHMV process compares favorably to state-of-the-art volatility models in terms of in-sample fit and out-of-sample forecasting performance over time horizons ranging from 1 to 100 days. Supplementary materials for this article are available online.     

A latent Markov model for detecting patterns of criminal activity

Summary.  The paper investigates the problem of determining patterns of criminal behaviour from official criminal histories, concentrating on the variety and type of offending convictions. The analysis is carried out on the basis of a multivariate latent Markov model which allows for discrete covariates affecting the initial and the transition probabilities of the latent process. We also show some simplifications which reduce the number of parameters substantially; we include a Rasch-like parameterization of the conditional distribution of the response variables given the latent process and a constraint of partial homogeneity of the latent Markov chain. For the maximum likelihood estimation of the model we outline an EM algorithm based on recursions known in the hidden Markov literature, which make the estimation feasible also when the number of time occasions is large. Through this model, we analyse the conviction histories of a cohort of offenders who were born in England and Wales in 1953. The final model identifies five latent classes and specifies common transition probabilities for males and females between 5-year age periods, but with different initial probabilities.     

A Dynamic Latent Model for Poverty Measurement

The main purpose of this paper is the longitudinal analysis of the poverty phenomenon. By interpreting poverty as a latent variable, we are able to resort to the statistical methodology developed for latent structure analysis. In particular, we propose to use the mixture latent Markov model which allows us to achieve two goals: (i) a time-invariant classification of households into homogenous groups, representing different levels of poverty; (ii) the dynamic analysis of the poverty phenomenon which highlights the distinction between transitory and permanent poverty situations. Furthermore, we exploit the flexibility provided by the model in order to achieve the measurement of poverty in a multidisciplinary framework, using several socio-economic indicators as covariates and identifying the main relevant factors which influence permanent and transitory poverty. The analysis of the longitudinal data of the Survey on Households Income and Wealth of the Bank of Italy provides the identification of two groups of households which are characterized by different dynamic features. Moreover, the inclusion of socio-economic covariates such as level of education, employment status, geographical area and residence size of the household head shows a direct association with permanent poverty.     

居民家庭金融资产风险与宏观经济波动的协动性关系研究——基于GARCH-M模型的风险度量方法

居民家庭金融资产组合不仅存在风险,而且风险会随着居民持有不同金融资产的比重、家庭外部宏观经济的波动而变动。通过构建GARCH-M模型,引入宏观经济变量(GDP增长率、CPI、利率),并计算在险价值VaR,分析宏观经济指标波动与家庭金融风险的协动性关系。研究认为,家庭金融收益主要受利率影响,家庭金融风险主要受GDP和CPI影响,而且家庭金融风险具有集聚性,并会影响利率的变化。     

Flexible Latent‐State Modelling of Old Faithful's Eruption Inter‐Arrival Times in 2009

This paper is concerned with the analysis of a time series comprising the eruption inter‐arrival times of the Old Faithful geyser in 2009. The series is much longer than other well‐documented ones and thus gives a more comprehensive insight into the dynamics of the geyser. Basic hidden Markov models with gamma state‐dependent distributions and several extensions are implemented. In order to better capture the stochastic dynamics exhibited by Old Faithful, the different non‐standard models under consideration seek to increase the flexibility of the basic models in various ways: (i) by allowing non‐geometric distributions for the times spent in the different states; (ii) by increasing the memory of the underlying Markov chain, with or without assuming additional structure implied by mixture transition distribution models; and (iii) by incorporating feedback from the observation process on the latent process. In each case it is shown how the likelihood can be formulated as a matrix product which can be conveniently maximized numerically.     

Perfect simulation of an inventory model for perishable products

We study an inventory model for perishable products with a critical-number ordering policy under the assumption that demand for the product forms an i.i.d. sequence, so that the state of the system forms a Markov chain. Explicit calculation of the stationary distribution has proved impractical in cases where items have reasonably long lifetimes and for systems with large under-up-to levels. Using the recently developed coupling-from-the-past method, we introduce a technique to estimate the stationary distribution of the Markov chain via perfect simulation. The Markov chain that results from the use of a critical-number policy is particularly amenable to these simulation techniques, despite not being ordered in its initial state, since the recursive equations satisfied by the Markov chain enable us to identify specific demand patterns where the backward coupling occurs.     

HEAVY-TAILED-DISTRIBUTED THRESHOLD STOCHASTIC VOLATILITY MODELS IN FINANCIAL TIME SERIES

To capture mean and variance asymmetries and time‐varying volatility in financial time series, we generalize the threshold stochastic volatility (THSV) model and incorporate a heavy‐tailed error distribution. Unlike existing stochastic volatility models, this model simultaneously accounts for uncertainty in the unobserved threshold value and in the time‐delay parameter. Self‐exciting and exogenous threshold variables are considered to investigate the impact of a number of market news variables on volatility changes. Adopting a Bayesian approach, we use Markov chain Monte Carlo methods to estimate all unknown parameters and latent variables. A simulation experiment demonstrates good estimation performance for reasonable sample sizes. In a study of two international financial market indices, we consider two variants of the generalized THSV model, with US market news as the threshold variable. Finally, we compare models using Bayesian forecasting in a value‐at‐risk (VaR) study. The results show that our proposed model can generate more accurate VaR forecasts than can standard models.     

Dynamic latent trait models with mixed hidden Markov structure for mixed longitudinal outcomes

We propose a general Bayesian joint modeling approach to model mixed longitudinal outcomes from the exponential family for taking into account any differential misclassification that may exist among categorical outcomes. Under this framework, outcomes observed without measurement error are related to latent trait variables through generalized linear mixed effect models. The misclassified outcomes are related to the latent class variables, which represent unobserved real states, using mixed hidden Markov models (MHMMs). In addition to enabling the estimation of parameters in prevalence, transition and misclassification probabilities, MHMMs capture cluster level heterogeneity. A transition modeling structure allows the latent trait and latent class variables to depend on observed predictors at the same time period and also on latent trait and latent class variables at previous time periods for each individual. Simulation studies are conducted to make comparisons with traditional models in order to illustrate the gains from the proposed approach. The new approach is applied to data from the Southern California Children Health Study to jointly model questionnaire-based asthma state and multiple lung function measurements in order to gain better insight about the underlying biological mechanism that governs the inter-relationship between asthma state and lung function development.     

Maximum likelihood estimation of bivariate circular hidden Markov models from incomplete data

In this paper, we propose a hidden Markov model for the analysis of the time series of bivariate circular observations, by assuming that the data are sampled from bivariate circular densities, whose parameters are driven by the evolution of a latent Markov chain. The model segments the data by accounting for redundancies due to correlations along time and across variables. A computationally feasible expectation maximization (EM) algorithm is provided for the maximum likelihood estimation of the model from incomplete data, by treating the missing values and the states of the latent chain as two different sources of incomplete information. Importance-sampling methods facilitate the computation of bootstrap standard errors of the estimates. The methodology is illustrated on a bivariate time series of wind and wave directions and compared with popular segmentation models for bivariate circular data, which ignore correlations across variables and/or along time.     

Statistical Inference for Partially Hidden Markov Models

ABSTRACT

In this article we introduce a new missing data model, based on a standard parametric Hidden Markov Model (HMM), for which information on the latent Markov chain is given since this one reaches a fixed state (and until it leaves this state). We study, under mild conditions, the consistency and asymptotic normality of the maximum likelihood estimator. We point out also that the underlying Markov chain does not need to be ergodic, and that identifiability of the model is not tractable in a simple way (unlike standard HMMs), but can be studied using various technical arguments.     

Mixture Multi-state Markov Regression Model

Although heterogeneity across individuals may be reduced when a two-state process is extended into a multi-state process, the discrepancy between the observed and the predicted for some states may still exist owing to two possibilities, unobserved mixture distribution in the initial state and the effect of measured covariates on subsequent multi-state disease progression. In the present study, we developed a mixture Markov exponential regression model to take account of the above-mentioned heterogeneity across individuals (subject-to-subject variability) with a systematic model selection based on the likelihood ratio test. The model was successfully demonstrated by an empirical example on surveillance of patients with small hepatocellular carcinoma treated by non-surgical methods. The estimated results suggested that the model with the incorporation of unobserved mixture distribution behaves better than the one without. Complete and partial effects regarding risk factors on different subsequent multi-state transitions were identified using a homogeneous Markov model. The combination of both initial mixture distribution and homogeneous Markov exponential regression model makes a significant contribution to reducing heterogeneity across individuals and over time for disease progression.     

On-Line Learning for the Infinite Hidden Markov Model

We develop a sequential Monte Carlo algorithm for the infinite hidden Markov model (iHMM) that allows us to perform on-line inferences on both system states and structural (static) parameters. The algorithm described here provides a natural alternative to Markov chain Monte Carlo samplers previously developed for the iHMM, and is particularly helpful in applications where data is collected sequentially and model parameters need to be continuously updated. We illustrate our approach in the context of both a simulation study and a financial application.     

Model-based segmentation of spatial cylindrical data

ABSTRACT

A new hidden Markov random field model is proposed for the analysis of cylindrical spatial series, i.e. bivariate spatial series of intensities and angles. It allows us to segment cylindrical spatial series according to a finite number of latent classes that represent the conditional distributions of the data under specific environmental conditions. The model parsimoniously accommodates circular–linear correlation, multimodality, skewness and spatial autocorrelation. A numerically tractable expectation–maximization algorithm is provided to compute parameter estimates by exploiting a mean-field approximation of the complete-data log-likelihood function. These methods are illustrated on a case study of marine currents in the Adriatic sea.     

Semiparametric latent variable regression models for spatiotemporal modelling of mobile source particles in the greater Boston area

Summary.  Traffic particle concentrations show considerable spatial variability within a metropolitan area. We consider latent variable semiparametric regression models for modelling the spatial and temporal variability of black carbon and elemental carbon concentrations in the greater Boston area. Measurements of these pollutants, which are markers of traffic particles, were obtained from several individual exposure studies that were conducted at specific household locations as well as 15 ambient monitoring sites in the area. The models allow for both flexible non-linear effects of covariates and for unexplained spatial and temporal variability in exposure. In addition, the different individual exposure studies recorded different surrogates of traffic particles, with some recording only outdoor concentrations of black or elemental carbon, some recording indoor concentrations of black carbon and others recording both indoor and outdoor concentrations of black carbon. A joint model for outdoor and indoor exposure that specifies a spatially varying latent variable provides greater spatial coverage in the area of interest. We propose a penalized spline formulation of the model that relates to generalized kriging of the latent traffic pollution variable and leads to a natural Bayesian Markov chain Monte Carlo algorithm for model fitting. We propose methods that allow us to control the degrees of freedom of the smoother in a Bayesian framework. Finally, we present results from an analysis that applies the model to data from summer and winter separately.     

A dynamic analysis of stock markets using a hidden Markov model

This paper proposes a framework to detect financial crises, pinpoint the end of a crisis in stock markets and support investment decision-making processes. This proposal is based on a hidden Markov model (HMM) and allows for a specific focus on conditional mean returns. By analysing weekly changes in the US stock market indexes over a period of 20 years, this study obtains an accurate detection of stable and turmoil periods and a probabilistic measure of switching between different stock market conditions. The results contribute to the discussion of the capabilities of Markov-switching models of analysing stock market behaviour. In particular, we find evidence that HMM outperforms threshold GARCH model with Student-t innovations both in-sample and out-of-sample, giving financial operators some appealing investment strategies.     

Deprivation analysis based on Bayesian latent class models

This article seeks to measure deprivation among Portuguese households, taking into account four well-being dimensions – housing, durable goods, economic strain and social relationships – with survey data from the European Community Household Panel. We propose a multi-stage approach to a cross-sectional analysis, side-stepping the sparse nature of the contingency tables caused by the large number of variables considered and bringing together partial and overall analyses of deprivation that are based on Bayesian latent class models via Markov Chain Monte Carlo methods. The outcomes demonstrate that there was a substantial improvement on household overall well-being between 1995 and 2001. The dimensions that most contributed to the risk of household deprivation were found to be economic strain and social relationships.     

An online estimation scheme for a Hull–White model with HMM-driven parameters

This paper considers the implementation of a mean-reverting interest rate model with Markov-modulated parameters. Hidden Markov model filtering techniques in Elliott (1994, Automatica, 30:1399–1408) and Elliott et al. (1995, Hidden Markov Models: Estimation and Control. Springer, New York) are employed to obtain optimal estimates of the model parameters via recursive filters of auxiliary quantities of the observation process. Algorithms are developed and implemented on a financial dataset of 30-day Canadian Treasury bill yields. We also provide standard errors for the model parameter estimates. Our analysis shows that within the dataset and period studied, a model with two regimes is sufficient to describe the interest rate dynamics on the basis of very small prediction errors and the Akaike information criterion.     

Model-based clustering for social networks

Summary.  Network models are widely used to represent relations between interacting units or actors. Network data often exhibit transitivity, meaning that two actors that have ties to a third actor are more likely to be tied than actors that do not, homophily by attributes of the actors or dyads, and clustering. Interest often focuses on finding clusters of actors or ties, and the number of groups in the data is typically unknown. We propose a new model, the latent position cluster model , under which the probability of a tie between two actors depends on the distance between them in an unobserved Euclidean 'social space', and the actors' locations in the latent social space arise from a mixture of distributions, each corresponding to a cluster. We propose two estimation methods: a two-stage maximum likelihood method and a fully Bayesian method that uses Markov chain Monte Carlo sampling. The former is quicker and simpler, but the latter performs better. We also propose a Bayesian way of determining the number of clusters that are present by using approximate conditional Bayes factors. Our model represents transitivity, homophily by attributes and clustering simultaneously and does not require the number of clusters to be known. The model makes it easy to simulate realistic networks with clustering, which are potentially useful as inputs to models of more complex systems of which the network is part, such as epidemic models of infectious disease. We apply the model to two networks of social relations. A free software package in the R statistical language, latentnet, is available to analyse data by using the model.     

Comparison of Selected Methods for Modeling of Multi-State Disease Progression Processes: A Simulation Study

Prognostic studies are essential to understand the role of particular prognostic factors and, thus, improve prognosis. In most studies, disease progression trajectories of individual patients may end up with one of mutually exclusive endpoints or can involve a sequence of different events.

One challenge in such studies concerns separating the effects of putative prognostic factors on these different endpoints and testing the differences between these effects.

In this article, we systematically evaluate and compare, through simulations, the performance of three alternative multivariable regression approaches in analyzing competing risks and multiple-event longitudinal data. The three approaches are: (1) fitting separate event-specific Cox's proportional hazards models; (2) the extension of Cox's model to competing risks proposed by Lunn and McNeil; and (3) Markov multi-state model.

The simulation design is based on a prognostic study of cancer progression, and several simulated scenarios help investigate different methodological issues relevant to the modeling of multiple-event processes of disease progression. The results highlight some practically important issues. Specifically, the decreased precision of the observed timing of intermediary (non fatal) events has a strong negative impact on the accuracy of regression coefficients estimated with either the Cox's or Lunn-McNeil models, while the Markov model appears to be quite robust, under the same circumstances. Furthermore, the tests based on both Markov and Lunn-McNeil models had similar power for detecting a difference between the effects of the same covariate on the hazards of two mutually exclusive events. The Markov approach yields also accurate Type I error rate and good empirical power for testing the hypothesis that the effect of a prognostic factor on changes after an intermediary event, which cannot be directly tested with the Lunn-McNeil method. Bootstrap-based standard errors improve the coverage rates for Markov model estimates. Overall, the results of our simulations validate Markov multi-state model for a wide range of data structures encountered in prognostic studies of disease progression, and may guide end users regarding the choice of model(s) most appropriate for their specific application.