This paper studies the estimation of a class of copula-based semiparametric stationary Markov models. These models are characterized by nonparametric invariant (or marginal) distributions and parametric copula functions that capture the temporal dependence of the processes; the implied transition distributions are all semiparametric. Models in this class are easy to simulate, and can be expressed as semiparametric regression transformation models. One advantage of this copula approach is to separate out the temporal dependence (such as tail dependence) from the marginal behavior (such as fat tailedness) of a time series. We present conditions under which processes generated by models in this class are β-mixing; naturally, these conditions depend only on the copula specification. Simple estimators of the marginal distribution and the copula parameter are provided, and their asymptotic properties are established under easily verifiable conditions. Estimators of important features of the transition distribution such as the (nonlinear) conditional moments and conditional quantiles are easily obtained from estimators of the marginal distribution and the copula parameter; their √ n− consistency and asymptotic normality can be obtained using the Delta method. In addition, the semiparametric conditional quantile estimators are automatically monotonic across quantiles.JEL Classification: C14; C22

and Toulouse. We also thank R. Lestringand who performed the numerical illustration given in Section 5. 2 Lyxor Asset Management and CREST.

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We introduce a new class of semiparametric copula-based multivariate dynamic (SCOMDY) models, which specify the conditional mean and the conditional variance of a multivariate time series parametrically, but specify the multivariate distribution of the standardized innovation semiparametrically as a parametric copula evaluated at nonparametric marginal distributions. We first study large sample properties of the estimators of SCOMDY model parameters under a misspecified parametric copula, then propose pseudo likelihood ratio (PLR) tests for model selection between two SCOMDY models with possibly misspecified copulas, and finally develop PLR tests for model selection between more than two SCOMDY models. The limiting null distributions of the PLR tests do not depend on the estimation of conditional mean and conditional variance parameters, hence are very easy to simulate. Empirical applications to three and higher dimensional daily exchange rate series indicate that a SCOMDY model with a tail-dependent copula is generally preferred. r

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Fan et al. Reply: The development of atomistic methods capable of following microstructural evolution over time scales beyond the reach of traditional molecular dynamics simulations is a continuing challenge. We offer two observations concerning the preceding Comment [1] regarding the accuracy of the autonomous basin climbing (ABC) method and the benchmarking of new methods such as ABC and the kinetic activation-relaxation technique (k-ART). First, our experience with ABC suggests that it is a simple and robust algorithm for escaping from the potential energy surface (PES) minima in providing the dominant transition pathway [2][3][4][5][6]. In the present published form of ABC, however, it is likely to give an overestimation of effective transition time because of the 1D nature of sampling and implementation. We note, on the other hand, that the ABC method is not inherently limited to producing only a 1D trajectory for the system evolution. To give an illustration we consider a synthetic comparison between ABC and kinetic Monte Carlo (KMC) methods using a preconstructed two-dimensional PES [ Fig. 1(a)]. The ABC trajectory was generated for the same prescribed initial state, and used to estimate the temperature-dependent system evolution time and the effective activation energy. Six different sets of activation penalty function parameters [2] were employed, and all the runs gave the same governing transition pathway. To simulate a nonequilibrium driving force that resembles the vacancy clustering problem (see Fig. 2 Fig. 1(a)] to form a rough energy landscape, as well as a forward bias connecting the initial and final states. As shown in Fig. 1(b), ABC gives the same effective barrier (1.28 eV) as KMC simulations, while overestimating the evolution times by about 1-2 orders of magnitude. We believe this is a result of the 1D nature of the sampled trajectories in the published version of ABC rather than missing the dominant transition pathways. That is, the residence time at each state is governed by 1=ðk forward þ k backward Þ, whereas in KMC simulations it is governed by 1= P k i , considering all possible pathways i. This suggests that the evolution time in the vacancy clustering problem [7] could have been overestimated to some extent; however, it does not seem reasonable that the difference in sampling effectiveness alone can account for evolution times differing by 8 orders of magnitude [1,7]. Moreover, we are currently extending ABC to sample multiple transition pathways, akin to onthe-fly KMC simulations, by blocking the explored transitions. The ABC method, inherently due to its algorithm, favors finding the lowest-energy activation paths. Therefore, this new procedure can efficiently construct the event catalog from highly likely events to lowprobability events. As expected, the results from this extended version of the method (ABC-E) show a higher numerical accuracy, as demonstrated for the same 2D PES in Fig. 1. Second, the usefulness of an atomistic method in producing a minimum-energy path for the ...

The studies on dynamics and deformation in glassy materials are particularly challenging because of their strongly disordered atomic structure. Here, by probing the changes in the atomic displacements and stresses at saddle points of the potential energy landscape, we show that thermally activated deformation is triggered by subnano-scale rearrangements of a small number of atoms, typically less than 10 atoms. The individual triggers are invariant of the cooling history or elastic structure of the system. However, the organizations between different trigger centres can be varied and are related to the overall stability of the system. This finding allows a semi-quantitative construction of the potential energy landscape and brings a new perspective to the study of the mechanical properties of glasses.

Let F denote a distribution function defined on the probability space (Ω,,P), which is absolutely continuous with respect to the Lebesgue measure in Rd with probability density function f. Let f0(·,β) be a parametric density function that depends on an unknown p × 1 vector β. In this paper, we consider tests of the goodness-of-fit of f0(·,β) for f(·) for some β based on (i) the integrated squared difference between a kernel estimate of f(·) and the quasimaximum likelihood estimate of f0(·,β) denoted by In and (ii) the integrated squared difference between a kernel estimate of f(·) and the corresponding kernel smoothed estimate of f0(·, β) denoted by Jn. It is shown in this paper that the amount of smoothing applied to the data in constructing the kernel estimate of f(·) determines the form of the test statistic based on In. For each test developed, we also examine its asymptotic properties including consistency and the local power property. In particular, we show that tests developed in this paper, except the first one, are more powerful than the Kolmogorov-Smirnov test under the sequence of local alternatives introduced in Rosenblatt [12], although they are less powerful than the Kolmogorov-Smirnov test under the sequence of Pitman alternatives. A small simulation study is carried out to examine the finite sample performance of one of these tests.