Inference for a Special Bilinear Time Series Model

It is well known that estimating bilinear models is quite challenging. Many different ideas have been proposed to solve this problem. However, there is not a simple way to do inference even for its simple cases. This paper studies the special bilinear model $$Y_t=\mu+\phi Y_{t-2}+ bY_{t-2}\varepsilon_{t-1}+ \varepsilon_t,$$ where $\{\varepsilon_t\}$ is a sequence of i.i.d. random variables with mean zero. We first give a sufficient condition for the existence of a unique stationary solution for the model and then propose a GARCH-type maximum likelihood estimator for estimating the unknown parameters. It is shown that the GMLE is consistent and asymptotically normal under only finite fourth moment of errors. Also a simple consistent estimator for the asymptotic covariance is provided. A simulation study confirms the good finite sample performance. Our estimation approach is novel and nonstandard and it may provide a new insight for future research in this direction.Comment: 23 pages, 1 figures, 3 table

A survey of Hirota's difference equations

A review of selected topics in Hirota's bilinear difference equation (HBDE) is given. This famous 3-dimensional difference equation is known to provide a canonical integrable discretization for most important types of soliton equations. Similarly to the continuous theory, HBDE is a member of an infinite hierarchy. The central point of our exposition is a discrete version of the zero curvature condition explicitly written in the form of discrete Zakharov-Shabat equations for M-operators realized as difference or pseudo-difference operators. A unified approach to various types of M-operators and zero curvature representations is suggested. Different reductions of HBDE to 2-dimensional equations are considered. Among them discrete counterparts of the KdV, sine-Gordon, Toda chain, relativistic Toda chain and other typical examples are discussed in detail.Comment: LaTeX, 43 pages, LaTeX figures (with emlines2.sty

Notes on the BMS group in three dimensions: I. Induced representations

The Bondi-Metzner-Sachs group in three dimensions is the symmetry group of asymptotically flat three-dimensional spacetimes. It is the semi-direct product of the diffeomorphism group of the circle with the space of its adjoint representation, embedded as an abelian normal subgroup. The structure of the group suggests to study induced representations; we show here that they are associated with the well-known coadjoint orbits of the Virasoro group and provide explicit representations in terms of one-particle states.Comment: 33 pages, LaTeX file. v3: Minimal changes in the introduction. Version published in JHE

A class of stochastic unit-root bilinear processes: Mixing properties and unit-root test

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Estimating Functions and Equations: An Essay on Historical Developments with Applications to Econometrics

The idea of using estimating functions goes a long way back, at least to Karl Pearson's introduction to the method of moments in 1894. It is now a very active area of research in the statistics literature. One aim of this chapter is to provide an account of the developments relating to the theory of estimating functions. Starting from the simple case of a single parameter under independence, we cover the multiparameter, presence of nuisance parameters and dependent data cases. Application of the estimating functions technique to econometrics is still at its infancy. However, we illustrate how this estimation approach could be used in a number of time series models, such as random coefficient, threshold, bilinear, autoregressive conditional heteroscedasticity models, in models of spatial and longitudinal data, and median regression analysis. The chapter is concluded with some remarks on the place of estimating functions in the history of estimation.

Introduction to the nonequilibrium functional renormalization group

In these lectures we introduce the functional renormalization group out of equilibrium. While in thermal equilibrium typically a Euclidean formulation is adequate, nonequilibrium properties require real-time descriptions. For quantum systems specified by a given density matrix at initial time, a generating functional for real-time correlation functions can be written down using the Schwinger-Keldysh closed time path. This can be used to construct a nonequilibrium functional renormalization group along similar lines as for Euclidean field theories in thermal equilibrium. Important differences include the absence of a fluctuation-dissipation relation for general out-of-equilibrium situations. The nonequilibrium renormalization group takes on a particularly simple form at a fixed point, where the corresponding scale-invariant system becomes independent of the details of the initial density matrix. We discuss some basic examples, for which we derive a hierarchy of fixed point solutions with increasing complexity from vacuum and thermal equilibrium to nonequilibrium. The latter solutions are then associated to the phenomenon of turbulence in quantum field theory.Comment: Lectures given at the 49th Schladming Winter School `Physics at all scales: The Renormalization Group' (to appear in the proceedings); 24 pages, 3 figure