Nonparametric model validations for hidden Markov models with applications in financial econometrics

We address the nonparametric model validation problem for hidden Markov models with partially observable variables and hidden states. We achieve this goal by constructing a nonparametric simultaneous confidence envelope for transition density function of the observable variables and checking whether the parametric density estimate is contained within such an envelope. Our specification test procedure is motivated by a functional connection between the transition density of the observable variables and the Markov transition kernel of the hidden states. Our approach is applicable for continuous-time diffusion models, stochastic volatility models, nonlinear time series models, and models with market microstructure noise.     

Inference on Self‐Exciting Jumps in Prices and Volatility Using High‐Frequency Measures

Dynamic jumps in the price and volatility of an asset are modelled using a joint Hawkes process in conjunction with a bivariate jump diffusion. A state‐space representation is used to link observed returns, plus nonparametric measures of integrated volatility and price jumps, to the specified model components, with Bayesian inference conducted using a Markov chain Monte Carlo algorithm. An evaluation of marginal likelihoods for the proposed model relative to a large number of alternative models, including some that have featured in the literature, is provided. An extensive empirical investigation is undertaken using data on the S&P 500 market index over the 1996–2014 period, with substantial support for dynamic jump intensities—including in terms of predictive accuracy—documented. Copyright © 2016 John Wiley & Sons, Ltd.     

On Markov games

In the paper it is demonstrated, how a dynamic programming approach may be useful for the analysis of Markov games or stochastic games. Markov games with finitely many stages are dealt with extensively. The existence of optimal Markov strategies is proven for finite stage Markov games using a shortcut of a proof by D erman for the analogous result for Markov decision processes. For Markov games with a countably infinite number of stages some results are summarized. Here again the results and the methods of proof have much in common with results and proofs for Markov decision processes. Actually the theory of Markov games is a generalisation. The paper contains short introductions into the theories of matrix games and tree games.     

Local GMM estimation of time series models with conditional moment restrictions

This paper investigates statistical properties of the local generalized method of moments (LGMM) estimator for some time series models defined by conditional moment restrictions. First, we consider Markov processes with possible conditional heteroskedasticity of unknown forms and establish the consistency, asymptotic normality, and semi-parametric efficiency of the LGMM estimator. Second, we undertake a higher-order asymptotic expansion and demonstrate that the LGMM estimator possesses some appealing bias reduction properties for positively autocorrelated processes. Our analysis of the asymptotic expansion of the LGMM estimator reveals an interesting contrast with the OLS estimator that helps to shed light on the nature of the bias correction performed by the LGMM estimator. The practical importance of these findings is evaluated in terms of a bond and option pricing exercise based on a diffusion model for spot interest rate.     

Latent Markov Modelling of Recidivism Data

This article discusses the application of latent Markov modelling for the analysis of recidivism data. We briefly examine the relations of Markov modelling with log–linear analysis, pointing out pertinent differences as well. We show how the restrictive Markov model may be more easily applicable by adding latent variables to the model, in which case the latent Markov model is a dynamic version of the latent class model. As an illustration, we apply latent Markov analysis on an empirical data set of juvenile prosecution careers, showing how the Markov analyses producing well-fitting and interpretable solutions. We end by comparing the possible contributions of Markov modelling in recidivism research, outlining its drawbacks as well. Recommendations and directions for future research conclude the article.     

齐次及非齐次马氏域变模型在股价泡沫检验中的应用

本文建立非齐次马氏域变模型检验股市价格泡沫,并结合对1996年1月至2010年6月间我国股市价格泡沫程度的实证度量,与齐次马氏域变模型的检验效果进行比较。两模型都能在一定程度上反映我国股票市场价格在各时间段的不同泡沫水平,但非齐次马氏域变模型所得到有泡沫概率和无泡沫概率区分度更高,比齐次马氏域变模型能更加精确的区分各时点上的泡沫水平。实证结果也表明,近期我国应对金融危机的超宽松经济政策对资本市场的刺激作用较为直接,而对实体经济的作用相对滞后,股市泡沫水平再度上升。     

灰色-马尔可夫链模型在股市预测中的应用

用GM(1,1)预测具有良好的精确性和规律性,但对于随机波动性较大的股市行业,它的预测精度比较低,而马尔可夫模型可以克服波动性较大的局限性,弥补灰色模型的不足,因此将两者结合起来对股市进行预测将能提高预测的精度。本文依据上交所20个月末收盘指数预测后四个月的月末收盘指数范围,实证分析表明灰色马尔可夫链模型在股市预测中应用的可行性     

Exit problems in regime-switching models

This paper provides a general framework for pricing of perpetual American and real options in regime-switching Lévy models. In each state of the Markov chain, which determines switches from one Lévy process to another, the payoff stream is a monotone function of the Lévy process labeled by the state. This allows for additional switching within each state of the Markov chain (payoffs can be different in different regions of the real line). The pricing procedure is efficient even if the number of states is large provided the transition rates are not very large w.r.t. the riskless rates. The payoffs and riskless rates may depend on a state. Special cases are stochastic volatility models and models with stochastic interest rate; both must be modeled as finite-state Markov chains. As an application, we solve exit problems for a price-taking firm, and study the dependence of the exit threshold on the interest rate uncertainty.     

Increasing risk: Dynamic mean-preserving spreads

We extend the celebrated Rothschild and Stiglitz (1970) definition of Mean-Preserving Spreads to a dynamic framework. We adapt the original integral conditions to transition probability densities, and give sufficient conditions for their satisfaction. We then focus on a class of nonlinear scalar diffusion processes, the super-diffusive ballistic process, and prove that it satisfies the integral conditions. We further prove that this class is unique among Brownian bridges. This class of processes can be generated by a random superposition of linear Markov processes with constant drifts. This exceptionally simple representation enables us to systematically revisit, by means of the properties of dynamic mean-preserving spreads, workhorse economic models originally based on White Gaussian Noise. A selection of four examples is presented and explicitly solved.     

证券投资组合风险管理的波动状态转移分析

运用马尔科夫区制转移模型对我国股票型开放式基金收益率波动进行了风险评估和短期预测。采用晨星评级净值收益率前10只股票型基金跨期四年共816日交易数据,进行风险价值计量;并将计量结果与市场贝塔系数模型进行比较分析。通过失败比例测试证实了波动状态转移风险控制模型的风险度量与控制能力显著优于市场贝塔系数模型,丰富与提升了资本资产定价理论的内涵,一定程度上提高了我国投资组合风险波动的控制、拟合和预测的精确性。     

Dating Business Cycles: A Methodological Contribution with an Application to the Euro Area

This paper proposes a dating algorithm based on an appropriately defined Markov chain that enforces alternation of peaks and troughs, and duration constraints concerning the phases and the full cycle. The algorithm, which implements Harding and Pagan's non‐parametric dating methodology, allows an assessment of the uncertainty of the estimated turning points caused by filtering and can be used to construct indices of business cycle diffusion, aiming at assessing how widespread are cyclical movements throughout the economy. Its adaptation to the notion of a deviation cycle and the imposition of depth constraints are also discussed. We illustrate the algorithm with reference to the issue of dating the euro‐area business cycle and analysing its characteristics, both from the classical and the growth cycle perspectives.     

马尔可夫性及其检验方法研究

马尔可夫链广泛应用于信息论、自动控制、通信技术、基因遗传、计算机科学、经济管理、教育管理、市场预测等领域,马尔可夫性是其最基本特征,因此,在运用马尔可夫链进行预测时,必须对预测对象以往的统计数据资料构成的随机变量序列的马尔可夫性进行检验,只有符合马尔可夫性,才能利用马尔可夫链进行预测,才能保证预测精度。本文探讨了马氏性的概念及性质,研究了马尔可夫性的检验方法,通过实例对该检验方法进行了分析。     

Testing the Markov property with high frequency data

This paper develops a framework to nonparametrically test whether discrete-valued irregularly spaced financial transactions data follow a Markov process. For that purpose, we consider a specific optional sampling in which a continuous-time Markov process is observed only when it crosses some discrete level. This framework is convenient for it accommodates the irregular spacing that characterizes transactions data. Under such an observation rule, the current price duration is independent of a previous price duration given the previous price realization. A simple nonparametric test then follows by examining whether this conditional independence property holds. Monte Carlo simulations suggest that the asymptotic test has huge size distortions, though a bootstrap-based variant entails reasonable size and power properties in finite samples. As for an empirical illustration, we investigate whether bid–ask spreads follow Markov processes using transactions data from the New York Stock Exchange. The motivation lies on the fact that asymmetric information models of market microstructures predict that the Markov property does not hold for the bid–ask spread. We robustly reject the Markov assumption for two out of the five stocks under scrutiny. Finally, it is reassuring that our results are consistent with two alternative measures of asymmetric information.     

Coefficients of ergodicity for Markov chains with uncertain parameters

One of the central considerations in the theory of Markov chains is their convergence to an equilibrium. Coefficients of ergodicity provide an efficient method for such an analysis. Besides giving sufficient and sometimes necessary conditions for convergence, they additionally measure its rate. In this paper we explore coefficients of ergodicity for the case of imprecise Markov chains. The latter provide a convenient way of modelling dynamical systems where parameters are not determined precisely. In such cases a tool for measuring the rate of convergence is even more important than in the case of precisely determined Markov chains, since most of the existing methods of estimating the limit distributions are iterative. We define a new coefficient of ergodicity that provides necessary and sufficient conditions for convergence of the most commonly used class of imprecise Markov chains. This so-called weak coefficient of ergodicity is defined through an endowment of the structure of a metric space to the class of imprecise probabilities. Therefore we first make a detailed analysis of the metric properties of imprecise probabilities.     

Markov Perfect equilibria in repeated asynchronous choice games

This paper examines the issue of multiplicity of Markov Perfect equilibria in alternating move repeated games. Such games are canonical models of environments with repeated, asynchronous choices due to inertia or replacement. Our main result is that the number of Markov Perfect equilibria is generically finite with respect to stage game payoffs. This holds despite the fact that the stochastic game representation of the alternating move repeated game is “non-generic” in the larger space of state dependent payoffs. We further obtain that the set of completely mixed Markov Perfect equilibria is generically empty with respect to stage game payoffs.     

The application of a Markov Chain in educational planning

Mathematical models can assist educators in the preparation of their educational plans and their potential in this regard is being increasingly realized. As a result, models have found application at all levels at which planning is conducted.This paper examines the potential of one model—the Markov Chain—which is capable of predicting enrolments for an education system.The model is applied to the New South Wales State Government education system between 1947 and 1961 and the projected enrolments compared to the actual enrolments in those years.Some success is achieved but it appears as if the data rather than the model are responsible for this. The limitations of the Markov Chain approach are discussed and present research and directions listed.     

MC's for MCMC'ists

We develop a minimum amount of theory of Markov chains at as low a level of abstraction as possible in order to prove two fundamental probability laws for standard Markov chain Monte Carlo algorithms:
1. The law of large numbers explains why the algorithm works: it states that the empirical means calculated from the samples converge towards their "true" expected values, viz. expectations with respect to the invariant distribution of the associated Markov chain (=the target distribution of the simulation).
2. The central limit theorem expresses the deviations of the empirical means from their expected values in terms of asymptotically normally distributed random variables. We also present a formula and an estimator for the associated variance.     

Does the Macroeconomy Predict UK Asset Returns in a Nonlinear Fashion? Comprehensive Out‐of‐Sample Evidence

We perform a comprehensive examination of the recursive, comparative predictive performance of linear and nonlinear models for UK stock and bond returns. We estimate Markov switching, threshold autoregressive (TAR) and smooth transition autoregressive (STR) regime switching models and a range of linear specifications including models with GARCH type specifications. Results demonstrate UK asset returns require nonlinear dynamics to be modelled with strong evidence in favour of Markov switching frameworks. Our results appear robust to the choice of sample period, changes in loss functions and to the methodology employed to test for equal predictive accuracy. The key findings extend to a similar sample of US data.     

Interconnections Between Eurozone and US Booms and Busts Using a Bayesian Panel Markov‐Switching VAR Model

The proposed panel Markov‐switching VAR model accommodates changes in low and high data frequencies and incorporates endogenous time‐varying transition matrices of country‐specific Markov chains, allowing for interconnections. An efficient multi‐move sampling algorithm draws time‐varying Markov‐switching chains. Using industrial production growth and credit spread data, several important data features are obtained. Three regimes appear, with slow growth becoming persistent in the eurozone. Turning point analysis indicates the USA leading the eurozone cycle. Amplification effects influence recession probabilities for Eurozone countries. A credit shock results in temporary negative industrial production growth in Germany, Spain and the USA. Core and peripheral countries exist in the eurozone. Copyright © 2016 John Wiley & Sons, Ltd.     

A unified approach to Bermudan and barrier options under stochastic volatility models with jumps

Many financial assets, such as currencies, commodities, and equity stocks, exhibit both jumps and stochastic volatility, which are especially prominent in the market after the financial crisis. Some strategic decision making problems also involve American-style options. In this paper, we develop a novel, fast and accurate method for pricing American and barrier options in regime switching jump diffusion models. By blending regime switching models and Markov chain approximation techniques in the Fourier domain, we provide a unified approach to price Bermudan, American options and barrier options under general stochastic volatility models with jumps. The models considered include Heston, Hull–White, Stein–Stein, Scott, the 3/2 model, and the recently proposed 4/2 model and the α-Hypergeometric model with general jump amplitude distributions in the return process. Applications include the valuation of discretely monitored contracts as well as continuously monitored contracts common in the foreign exchange markets. Numerical results are provided to demonstrate the accuracy and efficiency of the proposed method.