Flexible Threshold Models for Modelling Interest Rate Volatility

This paper focuses on interest rate models with regime switching and extends previous nonlinear threshold models by relaxing the assumption of a fixed number of regimes. Instead we suggest automatic model determination through Bayesian inference via the reversible jump Markov Chain Monte Carlo (MCMC) algorithm. Moreover, we allow the thresholds in the volatility to be driven not only by the interest rate but also by other economic factors. We illustrate our methodology by applying it to interest rates and other economic factors of the American economy.     

Flexible Threshold Models for Modelling Interest Rate Volatility

This paper focuses on interest rate models with regime switching and extends previous nonlinear threshold models by relaxing the assumption of a fixed number of regimes. Instead we suggest automatic model determination through Bayesian inference via the reversible jump Markov Chain Monte Carlo (MCMC) algorithm. Moreover, we allow the thresholds in the volatility to be driven not only by the interest rate but also by other economic factors. We illustrate our methodology by applying it to interest rates and other economic factors of the American economy.     

因子模型门槛效应的LM和Wald检验及其统计性质研究

门槛因子模型可以有效地刻画高维度时间序列的共变特征和区制转换行为,具有良好的可解释性和预测能力。针对因子载荷矩阵存在的门槛效应,本文提出了拉格朗日乘子和沃尔德检验方法,并给出了渐近分布,相关结果表明以上检验统计量具有良好的大样本性质和有限样本表现。在实证部分,以我国股市的行业指数作为研究对象,通过构建门槛因子模型来刻画我国股票市场波动的共变性特征和非对称效应。实证结果表明基于门槛因子模型可以很好地刻画中国股市行业收益率波动的共变特征和区制转换行为。     

Models for Extremal Dependence Derived from Skew‐symmetric Families

Skew‐symmetric families of distributions such as the skew‐normal and skew‐t represent supersets of the normal and t distributions, and they exhibit richer classes of extremal behaviour. By defining a non‐stationary skew‐normal process, which allows the easy handling of positive definite, non‐stationary covariance functions, we derive a new family of max‐stable processes – the extremal skew‐t process. This process is a superset of non‐stationary processes that include the stationary extremal‐t processes. We provide the spectral representation and the resulting angular densities of the extremal skew‐t process and illustrate its practical implementation.     

Modeling rare events through a pRARMAX process

7 and 8 introduce a power max-autoregressive process, in short pARMAX, as an alternative to heavy tailed ARMA when modeling rare events. In this paper, an extension of pARMAX is considered, by including a random component which makes the model more applicable to real data. We will see conditions under which this new model, here denoted as pRARMAX, has unique stationary distribution and we analyze its extremal behavior. Based on Bortot and Tawn (1998), we derive a threshold-dependent extremal index which is a functional of the coefficient of tail dependence of 14 and 15 which in turn relates with the pRARMAX parameter. In order to fit a pRARMAX model to an observed data series, we present a methodology based on minimizing the Bayes risk in classification theory and analyze this procedure through a simulation study. We illustrate with an application to financial data.     

A transitional Markov switching autoregressive model

ABSTRACT

This paper is concerned with properties of a transitional Markov switching autoregressive (TMSAR) model, together with its maximum-likelihood estimation and inference. We extend existing MSAR models by allowing dependence of AR parameters on hidden states at time points prior to the current time t. A stationary solution is given and expressions for the theoretical autocovariance function are derived. Two time series are analyzed and the new model outperforms two existing MSAR models in terms of maximized log-likelihood, residual correlations, and one-step-ahead forecasting performance. The new model also gives more regime changes in agreement with real events.     

Tests for Unit-Root versus Threshold Specification With an Application to the Purchasing Power Parity Relationship

We consider modeling the real exchange rate by a stationary three-regime self-exciting threshold autoregressive (SETAR) model with possibly a unit root in the middle regime. This representation is consistent with purchasing power parity in the presence of trading costs. Our main contribution is to provide statistical tools for testing unit root versus a SETAR. First, we show that a SETAR with a unit root in the middle regime is stationary and mixing under reasonable assumptions. Second, we derive analytically the asymptotic distribution of our unit-root test under the null. Using monthly real exchange rate data, our test rejects the null of unit-root against a threshold process for five European series.     

Equivalence of Gaussian measures for some nonstationary random fields

Gaussian random fields whose covariance structures are described by a power law model provide a simple and flexible class of models for isotropic random fields. This class includes fractional Brownian fields as a special case. Because these random fields are nonstationary, the extensive results available on equivalence of Gaussian measures for stationary models do not apply to them. This work shows that results on equivalence for two stationary Gaussian random field models extend in a natural way to the equivalence of a stationary model and a power law model. This result is used to show that if we use a power law model for predicting a random field at unobserved locations when in fact the random field is stationary, we can obtain asymptotically optimal predictions as long as the high frequency behavior of the true spectral density is sufficiently close to the high frequency behavior of the spectral density of the power law model.     

Local Digital Algorithms Applied to Boolean Models

We investigate the estimation of specific intrinsic volumes of stationary Boolean models by local digital algorithms; that is, by weighted sums of local configuration counts. We show that asymptotically unbiased estimators for the specific surface area or integrated mean curvature do not exist if the dimension is at least two or three, respectively. For three‐dimensional stationary isotropic Boolean models, we derive asymptotically unbiased estimators for the specific surface area and integrated mean curvature. For a Boolean model with balls as grains, we even obtain an asymptotically unbiased estimator for the specific Euler characteristic.     

Markov-switching generalized additive models

We consider Markov-switching regression models, i.e. models for time series regression analyses where the functional relationship between covariates and response is subject to regime switching controlled by an unobservable Markov chain. Building on the powerful hidden Markov model machinery and the methods for penalized B-splines routinely used in regression analyses, we develop a framework for nonparametrically estimating the functional form of the effect of the covariates in such a regression model, assuming an additive structure of the predictor. The resulting class of Markov-switching generalized additive models is immensely flexible, and contains as special cases the common parametric Markov-switching regression models and also generalized additive and generalized linear models. The feasibility of the suggested maximum penalized likelihood approach is demonstrated by simulation. We further illustrate the approach using two real data applications, modelling (i) how sales data depend on advertising spending and (ii) how energy price in Spain depends on the Euro/Dollar exchange rate.     

A seasonal mixed-POT model to estimate high flood quantiles from different event types and seasons

Flood events can be caused by several different meteorological circumstances. For example, heavy rain events often lead to short flood events with high peaks, whereas snowmelt normally results in events of very long duration with a high volume. Both event types have to be considered in the design of flood protection systems. Unfortunately, all these different event types are often included in annual maximum series (AMS) leading to inhomogeneous samples. Moreover, certain event types are underrepresented in the AMS. This is especially unsatisfactory if the most extreme events result from such an event type. Therefore, monthly maximum data are used to enlarge the information spectrum on the different event types. Of course, not all events can be included in the flood statistics because not every monthly maximum can be declared as a flood. To take this into account, a mixture Peak-over-threshold model is applied, with thresholds specifying flood events of several types that occur in a season of the year. This model is then extended to cover the seasonal type of the data. The applicability is shown in a German case study, where the impact of the single event types in different parts of a year is evaluated.     

中国通货膨胀率的非线性动态行为研究——来自MS—ARFIMA模型的新证据

使用允许长记忆参数d服从区制转换的MS—ARFIMA模型对中国月度通货膨胀路径的动态行为进行新的实证研究,结果显示：中国通货膨胀不仅均值水平和不确定性存在着“低通胀”区制和“高通胀”区制,而且更为重要的是,通货膨胀序列的平稳性也表现出显著的区制转换动态。“低通胀”区制下,长记忆参数d1=0．361,说明通货膨胀是协方差平稳序列,“高通胀”区制下,长记忆参数d2=1．145,说明通货膨胀是非平稳序列。这一新的研究结论意味着中国通货膨胀冲击的持久性效应也存在相应的区制转移变化。这要求央行在管控通货膨胀过程中,既要考虑均值和不确定性的区制变化,又要兼顾平稳性和持久性的区制变化。     

Extremes of Markov chains with tail switching potential

Summary.  A recent advance in the utility of extreme value techniques has been the characteri- zation of the extremal behaviour of Markov chains. This has enabled the application of extreme value models to series whose temporal dependence is Markovian, subject to a limitation that prevents switching between extremely high and extremely low levels. For many applications this is sufficient, but for others, most notably in the field of finance, it is common to find series in which successive values switch between high and low levels. We term such series Markov chains with tail switching potential, and the scope of this paper is to generalize the previous theory to enable the characterization of the extremal properties of series displaying this type of behaviour. In addition to theoretical developments, a modelling procedure is proposed. A simulation study is made to assess the utility of the model in inferring the extremal dependence structure of autoregressive conditional heteroscedastic processes, which fall within the tail switching Markov family, and generalized autoregressive conditional heteroscedastic processes which do not, being non-Markov in general. Finally, the procedure is applied to model extremal aspects of a financial index extracted from the New York Stock Exchange compendium.     

Sampled autocovariance and autocorrelation results for linear time processes

We derive an exact formula for the covariance between the sampled autocovariances at any two lags for a finite time series realisation from a general stationary autoregressive moving average process. We indicate, through one particular example, how this result can be used to deduce analogous formulae for any nonstationary model of the ARUMA class, a generalisation of the ARIMA models. Such formulae then allow us to obtain approximate expressions for the convariances between all pairs of serial correlations for finite realisations from the ARUMA model. We also note that, in the limit as the series length n → ∞, our results for the ARMA class retrieve those of Bartlett (1946). Finally, we investigate an improvement to the approximation that is obtained by applying Bartlett's general asymptotic formula to finite series realisations. That such an improvement should exist can immediately be seen by consideration of out results for the simplest case of a white noise process. However, we deduce the final improved approapproximation, for general models, in two ways - from (corrected) results due to Davies and Newbold (1980), and by an alternative approach to theirs.     

THE BOTTOMLESS DAM WITH SEASONAL INPUTS

The stationary distribution of the depletion (from the maximum content) of a bottomless dam is derived for the case where the inputs and releases are seasonal. The method used is an extension of the method of Wishart for a problem in queueing theory (without involving seasonality). A numerical example is given illustrating the method for the case of two seasons.
In some recent work on dams (see Phatarfod, 1979; Pakes & Phatarfod, 1978), it was shown that it was more appropriate to consider the bottomless (or the infinitely deep) dam model rather than the topless (or the infinitely high) dam model as an approximation to the finite dam. The main consideration is that in real-life reservoir situations, the planned release or draft is almost always less than the mean inflow. It is obvious that with this condition the content of a topless dam will not have a stationary distribution, whereas if we consider the depletion of a dam from its top, this will have a stationary distribution. An additional reason for considering the bottomless dam model is that the model is mathematically more tractable; this property was noticed before for continuous time models (see e.g. Hasofer, 1966)     

On blocks and runs estimators of the extremal index

Given a sample from a stationary sequence of random variables, we study the blocks and runs estimators of the extremal index. Conditions are given for consistency and asymptotic normality of these estimators. We show that moment restrictions assumed by Hsing (Stochast. Process. Appl. 37(1), 117–139; Ann. Statist. 21(4), 2043-2021) may be relaxed if a stronger mixing condition holds. The CLT for the runs estimator seems to be proven for the first time.     

Reliability analysis for gap null gate based on model comparison criterion

The length of the gap is the key factor affecting its reliability. Based on the mechanism of the gap null gate, this paper regards the two endpoint thresholds of the gap length as bivariate random variables and establishes successful response models. Score test statistic is presented to test the correlation coefficient. The DIC criterion is also provided to compare the models. With the experimental data of the gap null gate, we build Probit model and Logit model as the successful response models, and prove that the correlation coefficients in the both models can be regarded as 0. By comparing the DIC value, we find that the Probit model is more suitable to describe the distribution of the endpoint thresholds of the reliability window. Finally, both the point estimation and interval estimation results of the reliability window are given to illustrate the feasibility of the method shown in the paper.     

A new class of models for bivariate joint tails

Summary.  A fundamental issue in applied multivariate extreme value analysis is modelling dependence within joint tail regions. The primary focus of this work is to extend the classical pseudopolar treatment of multivariate extremes to develop an asymptotically motivated representation of extremal dependence that also encompasses asymptotic independence. Starting with the usual mild bivariate regular variation assumptions that underpin the coefficient of tail dependence as a measure of extremal dependence, our main result is a characterization of the limiting structure of the joint survivor function in terms of an essentially arbitrary non-negative measure that must satisfy some mild constraints. We then construct parametric models from this new class and study in detail one example that accommodates asymptotic dependence, asymptotic independence and asymmetry within a straightforward parsimonious parameterization. We provide a fast simulation algorithm for this example and detail likelihood-based inference including tests for asymptotic dependence and symmetry which are useful for submodel selection. We illustrate this model by application to both simulated and real data. In contrast with the classical multivariate extreme value approach, which concentrates on the limiting distribution of normalized componentwise maxima, our framework focuses directly on the structure of the limiting joint survivor function and provides significant extensions of both the theoretical and the practical tools that are available for joint tail modelling.     

Estimating the probability of simultaneous rainfall extremes within a region: a spatial approach

In this paper we investigate the impact of model mis-specification, in terms of the dependence structure in the extremes of a spatial process, on the estimation of key quantities that are of interest to hydrologists and engineers. For example, it is often the case that severe flooding occurs as a result of the observation of rainfall extremes at several locations in a region simultaneously. Thus, practitioners might be interested in estimates of the joint exceedance probability of some high levels across these locations. It is likely that there will be spatial dependence present between the extremes, and this should be properly accounted for when estimating such probabilities. We compare the use of standard models from the geostatistics literature with max-stables models from extreme value theory. We find that, in some situations, using an incorrect spatial model for our extremes results in a significant under-estimation of these probabilities which – in flood defence terms – could lead to substantial under-protection.