Search

We present an estimation and forecasting method, based on a differential evolution MCMC method, for inference in GARCH models subjected to an unknown number of structural breaks at unknown dates. We treat break dates as parameters and determine the number of breaks by computing the marginal likelihoods of competing models. We allow for both recurrent and non-recurrent (change-point) regime specifications.We illustrate the estimation method through simulations and apply it to seven financial time series of daily returns. We find structural breaks in the volatility dynamics of all series and recurrent regimes in nearly all series. Finally, we carry out a forecasting exercise to evaluate the usefulness of structural break models.

Shock-initiated ignition is studied numerically for single step Arrhenius kinetics. This is relevant to hydrogen safety, because hydrogen detonates easily, and detonation in shocked mixture may occur in deflagration to detonation transition scenarios, due to shock reflections on obstacles subsequent to flame acceleration. Simulation of ignition behind a shock moving into combustible mixture is difficult because of the singular nature of the initial conditions. The solution method includes two components. First, space as an independent variable is replaced by the ratio space over time, and second, initial conditions at a small non-zero time are used, obtained in closed form from short time asymptotics. This way, the initial singularity is effectively removed and the early process is well resolved. This method was used to study how the leading shock strength, and the heat release, affects the shock-initiated ignition process. The Essentially Non-Oscillatory algorithm used captures ignition, hot spot growth, birth of a secondary shock and transition into a detonation. Results show that for weaker shock cases as well as for higher heat release the evolution is more rapid, that a secondary shock forms closer to the contact surface and quickly becomes a detonation wave.

An asymptotically consistent small Mach number model of a standing wave thermoacoustic engine has been developed. A simple thermoacoustic engine consists of a resonating tube within which is inserted an acoustically compact assembly, composed of a stack of conducting plates, placed between two heat exchangers. The model couples one-dimensional linear acoustics in the resonator with a low Mach number viscous and conducting flow in the stack/heat exchangers section. The latter is solved through a two-dimensional numerical simulation. Results show that the model successfully captures the dynamics of the starting process, at a much lower cost than a fully compressible simulation of the entire engine.

The structure of ZND waves under simple three step chain-branching kinetics is analyzed, assuming a slow initiation rate but arbitrary chain-branching activation energy. The analysis allows for a complete solution for the ZND wave in all cases, inside or outside the chain-branching explosion region, or close to the explosion limit. Results show that even when the von Neumann point is inside the explosion region, chain-branching effectively stops and the chain-branching radical concentration reaches a small near-steady value before all the reactant is consumed. Beyond that point, chemistry proceeds slowly, at a rate of the order of the initiation rate. For a von Neumann point relatively close to the limit, the reactant concentration is still quite significant when chain-branching stops, but diminishes for von Neumann points deeper inside the explosion region. The assumption that initiation is much slower than chain-branching is often quite accurate, in which case the length required for complete burn is orders of magnitude longer than the chain-branching length, so that as a practical matter, combustion never completes. In contrast, numerical simulation shows that under the same conditions, the cellular wave results in a more complete burn.

We estimate by Bayesian inference the mixed conditional heteroskedasticity model of Haas et al. (2004a Journal of Financial Econometrics 2, 211–50). We construct a Gibbs sampler algorithm to compute posterior and predictive densities. The number of mixture components is selected by the marginal likelihood criterion. We apply the model to the SP500 daily returns.

International audience; The influence of a resistive load on the starting performance of a standing-wave thermoacoustic engine is investigated numerically. The model used is based upon a low Mach number assumption; it couples the two-dimensional nonlinear flow and heat exchange within the thermoacoustic active cell with one-dimensional linear acoustics in the loaded resonator. For a given engine geometry, prescribed temperatures at the heat exchangers, prescribed mean pressure, and prescribed load, results from a simulation in the time domain include the evolution of the acoustic pressure in the active cell. That signal is then analyzed, extracting growth rate and frequency of the dominant modes. For a given load, the temperature difference between the two sides is then varied; the most unstable mode is identified and so is the corresponding critical temperature ratio between heater and cooler. Next, varying the load, a stability diagram is obtained, potentially with a predictive value. Results are compared with those derived from Rott�s linear theory as well as with experimental results found in the literature.

Markov-switching models are usually specified under the assumption that all the parameters change when a regime switch occurs. Relaxing this hypothesis and being able to detect which parameters evolve over time is relevant for interpreting the changes in the dynamics of the series, for specifying models parsimoniously, and may be helpful in forecasting. We propose the class of sticky infinite hidden Markov-switching autoregressive moving average models, in which we disentangle the break dynamics of the mean and the variance parameters. In this class, the number of regimes is possibly infinite and is determined when estimating the model, thus avoiding the need to set this number by a model choice criterion. We develop a new Markov chain Monte Carlo estimation method that solves the path dependence issue due to the moving average component. Empirical results on macroeconomic series illustrate that the proposed class of models dominates the model with fixed parameters in terms of point and density forecasts. Appendix available at: https://ssrn.com/abstract=2965668

We examine detonation waves with a four-step chain-branching reaction model that exhibits explosion limits close to the two lower limits of hydrogen–oxygen chemistry. The reaction model consists of a chain-initiation step and a chain-branching step, both temperature-dependent with Arrhenius kinetics, followed by two pressure-dependent termination steps. Increasing the chain-branching activation energy or the overdrive shortens the reaction length in the ZND wavelength and leads to more unstable detonations, according to multi-dimensional linear stability analysis. Corresponding numerical simulations show that detonations with weak chain-branching reactions have a wave structure similar to those with a single-step reaction; strong chain-branching detonations show distinct keystone features. Keystone regions are bounded by a discontinuity in reactivity across the shear layers emanating from the triple points at the intersection of the transverse waves and the main front. Especially in the strong case, chain...

We study multi-dimensional stability and perform high resolution two-dimensional numerical simulations of detonations with a four-step chain-branching reaction model. The reaction model is designed to approximate hydrogen chemistry. It consists of a chain-initiation step and a chain-branching step, both temperature-dependent with Arrhenius kinetics, followed by two pressure-dependent termination steps. Increasing the chain-branching activation energy shortens the ZND reaction length and leads to more unstable detonations, according to the stability analysis. Computations with four values of the chain-branching activation energy are performed both in narrow and wide channels. In the wider channel, all cases studied show distinct keystone-shaped regions, associated with substantial differences in reactivity across the shear layer hence of the time and distance until chain-branching takes place. As the chain-branching activation energy increases, cells take a shorter time to form, and the ratio cell length over width decreases. The cell size is dominated by the longer unstable wavelength even when high frequency modes are more unstable, but cells appear earlier in narrow channels than in wider ones. Initially, the cellular structure looks weaker, and the cell size is dominated by the shorter, more unstable wavelength, but eventually, it adapts to the longest unstable wavelength still consistent with the domain width.

We examine the multi-dimensional linear and non-linear stability of planar detonation waves for one-step Arrhenius rate with arbitrary reaction order. A normal mode analysis with an iterative shooting method is used in linear stability. Results show that the effect of decreasing the reaction order is roughly equivalent to increasing the activation energy, or decreasing the heat release and overdrive. Both the one- and two-dimensional linear stability spectra consist of a larger number of unstable modes at lower reaction order. Bifurcation to non-oscillatory modes occurs at slightly lower activation energy or higher heat release and overdrive for lower reaction order. Thus overall, a lower reaction order results in more unstable waves. Numerical simulations, using a Weighted Essentially Non-Oscillatory scheme, show that activation energy has a stronger effect on cell regularity than the reaction order. It has been suggested that a relationship may exist between transverse wavelength from multi-dimensional ...

Adaptive radial-based direction sampling (ARDS) algorithms are specified for Bayesian analysis of models with nonelliptical, possibly, multimodal target distributions. A key step is a radial-based transformation to directions and distances. After the transformations a Metropolis-Hastings method or, alternatively, an importance sampling method is applied to evaluate generated directions. Next, distances are generated from the exact target distribution by means of the numerical inverse transformation method. An adaptive procedure is applied to update the initial location and covariance matrix in order to sample directions in an efficient way. Tested on a set of canonical mixture models that feature multimodality, strong correlation, and skewness, the ARDS algorithms compare favourably with the standard Metropolis-Hastings and importance samplers in terms of flexibility and robustness. The empirical examples include a regression model with scale contamination and a mixture model for economic growth of the USA.

This paper presents a method capable of estimating richly parametrized versions of the dynamic conditional correlation (DCC) model that go beyond the standard scalar case. The algorithm is based on the maximization of a Gaussian quasi-likelihood using a Bregman-proximal trust-region method to handle the various non-linear stationarity and positivity constraints that arise in this context. We consider the general matrix Hadamard DCC model with full rank, rank equal to two and, additionally, two different rank one matrix specifications. In the last mentioned case, the elements of the vectors that determine the rank one parameter matrices are either arbitrary or parsimoniously defined using the Almon lag function. We use actual stock returns data in dimensions up to thirty in order to carry out performance comparisons according to several in- and out-of-sample criteria. Our empirical results show that the use of richly parametrized models adds value with respect to the conventional scalar case.

We present an estimation and forecasting method, based on a differential evolution MCMC method, for inference in GARCH models subjected to an unknown number of structural breaks at unknown dates. We treat break dates as parameters and determine the number of breaks by computing the marginal likelihoods of competing models. We allow for both recurrent and non-recurrent (change-point) regime specifications. We illustrate the estimation method through simulations and apply it to seven financial time series of daily returns. We find structural breaks in the volatility dynamics of all series and recurrent regimes in nearly all series. Finally, we carry out a forecasting exercise to evaluate the usefulness of structural break models.

Summary This paper explains how the Gibbs sampler can be used to perform Bayesian inference on GARCH models. Although the Gibbs sampler is usually based on the analytical knowledge of the full conditional posterior densities, such knowledge is not available in regression models with GARCH errors. We show that the Gibbs sampler can be combined with a unidimensional deterministic integration rule applied to each coordinate of the posterior density. The full conditional densities are evaluated and inverted numerically to obtain random draws of the joint posterior. The method is shown to be feasible and competitive compared with importance sampling and the Metropolis‐Hastings algorithm. It is applied to estimate an asymmetric Student‐GARCH model for the return on a stock exchange index, and to compute predictive option prices on the index. We prove, moreover, that a flat prior on the degrees of freedom parameter leads to an improper posterior density.

The structure of failed one-dimensional detonations is derived using a high activation energy analysis. On a time scale longer than the chemical time based upon the von Neumann temperature, high activation energy one-dimensional detonations break down into a weaker shock, a contact surface separating hot burned gases from a colder, unburned mixture, and an expansion wave. While this leading order solution is unaffected by chemistry, hence self-similar, the perturbation, accounting for the chemistry, which depends upon chemical times, is not. By accounting for the chemistry, the perturbation problem determines the delay until reignition of the detonation occurs. This problem is almost identical to the problem of initiation in the region between a shock and contact surface, which is created by the collision of two shock waves. The main difference between that problem and the current analysis is that the downstream boundary condition now consists of radiating acoustics into hot burned products, at the location of the surface discontinuity. When the Newtonian limit is applied, that is, for a ratio of the specific heats approaching unity, the hot spot at which reignition occurs approaches the location of the contact surface. The time and length to reignition are then found to vary exponentially with the activation energy of the mixture. However, the Newtonian limit is not a very realistic model, because it makes the interval between a Mach number of 1/√γ and 1 disappear: in this range of Mach numbers, adding energy to a steady flow lowers the temperature, hence the reaction rate.

Self-sustaining detonations exhibit a three-dimensional structure within the reaction zone which is much more complex than one-dimensional classical theory. While the structure of two-dimensional detonations is now fairly well understood, the details of the three-dimensional structure remain largely unknown. Numerical simulation, performed with the reactive Euler equations coupled to a single-step chemistry model, yields insight into the three-dimensional reaction zone structure. Simulations in a square channel show that, in the absence of wall losses, the three-dimensional transverse wave spacing is the same as in two dimensions. However, the transverse wave structure is much more intricate. Two perpendicular modes exist in this channel, and they are approximately one-quarter of a period out of phase. This phase shift accounts for the slapping wave phenomenon observed in experiments. The slapping wave imprints on smoke foils occur when the transverse wave collides with the wall. The curvature of the imprints is explained by the curvature of the slapping wave, which is convex toward the wall. The interaction of the two transverse waves results in a vorticity field that is much more complex than in two dimensions. In two dimensions, vorticity originates from a point singularity, which is now replaced by a line singularity that forms closed loops interconnecting the two sets of transverse waves. The interconnection changes after collisions, creating cuts and free edges on the sheet structure which otherwise remains globally interconnected. It is the vorticity in the part of the sheet initiating or ending along a free edge that rolls up into rings. The rings entrain unburned fluid behind the Mach stem, as in two dimensions. The vorticity system provides a strong coupling mechanism between the two orthogonal sets of transverse waves.

Induction time is an important measurement obtained in shock tube experiments, for use in calibration or validation of chemical kinetic schemes. Typically, these times are taken as representative of spatially uniform, constant volume combustion induction times. The actual process that happens in the shock tube is clearly more complex, however. In a first approximation, the flow might be described as being one-dimensional, inviscid and reactive, behind an incident or reflected shock, although in reality, multi-dimensional and viscous effects will play a role [1, 2]. The constant volume assumption may yield good results when dealing with highly diluted mixtures at relatively high post-shock temperatures [3]. However in other circumstances, such as more reactive mixtures at low post-shock temperatures, strong spatial pressure gradients may be present, potentially leading to significant uncertainties or inaccuracies. Assessing the actual accuracy of the constant volume assumption or alternatively improving the means used in validating kinetic schemes will require at the very least a one-dimensional simulation of the reacting flow in shocked mixture. Although the required simulations are one-dimensional, they are not straightforward, because the initial conditions are singular in the sense that at the instant when chemistry is triggered by the passage of the shock into the reactive mixture, there is no simulation domain containing reactive mixture, and that domain subsequently grows as the shock propagates. We have developed techniques that handle these difficulties, mainly in the context of the deflagration-to-detonation scenario [4]. The full unsteady problem is described by the reactive Euler’s equations, which are transformed from its original formulation, in which space x and time t are used as independent variables to η = x/t and t. This transformation effectively overcomes the non-existence of the initial physical domain. Chemistry is modeled using a three-step chain-branching mechanism originally proposed by Short & Quirk [5].

GARCH volatility models with fixed parameters are too restrictive for long time series due to breaks in the volatility process. Flexible alternatives are Markov-switching GARCH and change-point GARCH models. They require estimation by MCMC methods due to the path dependence problem. An unsolved issue is the computation of their marginal likelihood, which is essential for determining the number of regimes or change-points. We solve the problem by using particle MCMC, a technique proposed by Andrieu, Doucet, and Holenstein (2010). We examine the performance of this new method on simulated data, and we illustrate its use on several return series.

A model of an idealized thermoacoustic engine is formulated, coupling nonlinear flow and heat exchange in the heat exchangers and stack with a simple linear acoustic model of the resonator and load. Correct coupling results in an asymptotically consistent global model, in the small Mach number approximation. A well-resolved numerical solution is obtained for two-dimensional heat exchangers and stack. The model assumes that the heat exchangers and stack are shorter than the overall length by a factor of the order of a representative Mach number. The model is well-suited for simulation of the entire startup process, whereby as a result of some excitation, an initially specified temperature profile in the stack evolves toward a near-steady profile, eventually reaching stationary operation. A validation analysis is presented, together with results showing the early amplitude growth and approach of a stationary regime. Two types of initial excitation are used: Random noise and a small periodic wave. The set of assumptions made leads to a heat-exchanger section that acts as a source of volume but is transparent to pressure and to a local heat-exchanger model characterized by a dynamically incompressible flow to which a locally spatially uniform acoustic pressure fluctuation is superimposed.

We propose a new multivariate volatility model where the conditional distribution of a vector time series is given by a mixture of multivariate normal distributions. Each of these distributions is allowed to have a time-varying covariance matrix. The process can be globally covariance-stationary even though some components are not covariance-stationary. We derive some theoretical properties of the model such as the unconditional covariance matrix and autocorrelations of squared returns. The complexity of the model requires a powerful estimation algorithm. In a simulation study we compare estimation by maximum likelihood with the EM algorithm and Bayesian estimation with a Gibbs sampler. Finally, we apply the model to daily U.S. stock returns.

We consider the estimation of a large number of GARCH models, of the order of several hundreds. Our interest lies in the identification of common structures in the volatility dynamics of the univariate time series. To do so, we classify the series in an unknown number of clusters. Within a cluster, the series share the same model and the same parameters. Each cluster contains therefore similar series. We do not know a priori which series belongs to which cluster. The model is a finite mixture of distributions, where the component weights are unknown parameters and each component distribution has its own conditional mean and variance. Inference is done by the Bayesian approach, using data augmentation techniques. Simulations and an illustration using data on U.S. stocks are provided.

An asymptotically consistent regenerator model has been developed, assuming arbitrarily large viscous losses in the regenerator, taken as a porous medium with arbitrary topology. The theory uses the multiple scale method. It is based upon a scaling assumption whereby fluid displacement is of the order of the regenerator length. Other assumptions include a small Mach number assumption, and that the mesh size is small compared with lengths and diameters. A specific distinguished limit (i.e., a specific relationship between these two small parameters) is required for viscous losses to be of the same magnitude as the temporal pressure amplitude. This model is then applied to the entire refrigerator, with amplitudes now restricted to small values (still in a fully asymptotically consistent manner), and with an inertance tube represented by an acoustic model. Results are obtained, including pressure and temperature profiles along the regenerator, and overall performance for various inertance tube designs.

Experiments have shown that many relevant features characterizing detonation propagation can be correlated to the natural size that cells are experimentally observed to assume. But well-posedness of the cell-size concept is not so obvious from the standpoint of analysis and computational studies. It is only because of the chemical time that the problem cannot be reduced to a Riemann problem, without time or length scale; it is thus clear that any meaningful measure of the cell size is necessarily scaled by the chemical time. Still, this leaves plenty of room for discussion, and computations provide a convenient tool for further investigation. But until recently, even two-dimensional computations could only be performed in narrow domains, resulting in a cell width that adapts to the width of a domain, with, as a minimum, half a cell across the computational domain. We have now performed some wider computations, showing cells that become irregular, given enough width. There is then a range of sizes, so that the size assumed in wide domains cannot nnambiguously be used as an objective measure. Other alternate definitions, such as the minimm channel width that can support cells, all offer serious inconvenients and limitations. The only consistent way is then to define an average cell size, obtained by counting the number of irregular cells across and to divide the channel width by the number of cells. This criterion appears to yield consistent results. Finally, a grid resolution analysis shows that these results and conclusions remain unchanged as the grid is refined.

While the structure of oneand two-dimensional detonations is now fairly well-understood, from experiments and computations, the details of the threedimensional structure remain largely unknown. Indeed, the details of the three-dimensional reaction zone and wave structure behind the front cannot be resolved by experiments limited by diagnostics that cut along a plane of sight. For example, smoke foils only show twodimensional slices. Numerical simulation, performed with the reactive Euler equations coupled to a single-step chemistry model, yields insight into the threedimensional reaction zone structure. The threedimensional simulation considers a case, originally analyzed by Eipenbeck, which is roughly equivalent to a dilute oxygen-hydrogen-argon system, and which results in fairly regular cells, in a square channel and in the absence of wall losses. Results show a transverse wave structure much more intricate than in two dimensions, made up of two perpendicular modes phase-shifted by approximately one-quarter of a period, accounting for the slapping wave phenomenon observed in experiments. The slapping wave imprints on smoke foils occur when the transverse wave collides with the wall. The vorticity field forms closed loops interconnecting the two sets of transverse waves. Triple point line collisions alter the vortex sheet structure, creating cuts and free edges. But, because of the phase shift, the collisions in the two transverse directions are not simultaneous, and as a result, the cuts that occur with each collisions only affect one direction. The sheet structure remains connected along the other direction, so that, in contrast with the two-dimensional case, it remains globally interconnected.

Because of pressure gradients due to the flow across the machine, pressures are not equal in the expansion and the compression cylinder, and the indicator diagrams are not the same as if pressure were uniform. Pressure gradients result from the viscous stresses and from the inertia of the fluid. The effect of inertia can be divided into acoustics, which is unsteady, and acceleration of the flow, which is present even at time scales at which acoustics is insignificant. The only irreversibility directly associated with pressure gradients is the loss due to true viscous friction, while inertia effects are reversible. We analyze the magnitude of the various components to the pressure gradient; we discuss their effects and we idenm the different situations that may arise. We show that the energy equation can be written in a form consistent with the concept of a uniform stagnation enthalpy flux leaking through the regenerator. The viscous losses in the regenerator have little effect on the enthalpy flux, but they increase the entropy flux from expansion to compression space. As a result, while in each cylinder the heat flux balances the work flux and the regenerator enthalpy, the ratio of the expansion and compression work is no longer equal to the Carnot ratio. In contrast, while inertia of the fluid does sect the indicator diagrams, it has no effect on efficiency.

We performed a numerical study of the thermal and physical phenomena occurring in thermoacoustic wave generators. The goal of the simulation is to predict the amplification due to thermoacoustics of a wave initially of small amplitude. Therefore, we focus on the stack and the two heat exchangers, which we call the active cell, which is acoustically compact. The resonator area is split into two parts: the active cell, in which heat transfer takes place, and a resonator, in which the flow is acoustic. The flow in the two‐dimensional active cell can be approximated as a low Mach number viscous, conductive flow, subjected to spatially uniform pressure fluctuations. This model is formally derived using asymptotic expansions in terms of Mach number. The focus here is heat transfer between two successive stack plates. The two‐dimensional time‐dependent problem resulting from this model is solved numerically. Outside the active cell, flow in the resonator is described by a reversible acoustic one‐dimensional mode...

We have used the importance functions proposed in section III.2 with 5 models covering a variety of situations, as summarized in the following table