This paper first provides some useful results on a generalized random coefficient autoregressive model and a generalized hidden Markov model. These results simultaneously imply strict stationarity, existence of higher order moments, geometric ergodicity, and β-mixing with exponential decay rates, which are important properties for statistical inference. As applications, we then provide easy-to-verify sufficient conditions to ensure β-mixing and finite higher order moments for various linear and nonlinear GARCH(1,1), linear and power GARCH(p,q), stochastic volatility, and autoregressive conditional duration models. For many of these models, our sufficient conditions for existence of second moments and exponential β-mixing are also necessary. For several GARCH(1,1) models, our sufficient conditions for existence of higher order moments again coincide with the necessary ones in He and Terasvirta (1999, Journal of Econometrics 92, 173–192).

In other words, while the solvability criterion of Kϕ = r imposes r ∈ Ψ for some γ ≥ 1, we will get a geometric rate of decay of the regularization bias as soon as r ∈ Ψ for some γ > 1. Note that the upper bound β = 2 of the rate of decay is just a matter of normalization of the parameter α of regularization. By (3.19), ϕ − ϕ © cannot go to zero faster than [1 − q (α, λ)] ©

This paper proposes a version of the generalized method of moments procedure that handles both the case where the number of moment conditions is finite and the case where there is a continuum of moment conditions. Typically, the moment conditions are indexed by an index parameter that takes its values in an interval. The objective function to minimize is then the norm of the moment conditions in a Hilbert space. The estimator is shown to be consistent and asymptotically normal. The optimal estimator is obtained by minimizing the norm of the moment conditions in the reproducing kernel Hilbert space associated with the covariance. We show an easy way to calculate this estimator. Finally, we study properties of a specification test using overidentifying restrictions. Results of this paper are useful in many instances where a continuum of moment conditions arises. Examples include efficient estimation of continuous time regression models, cross-sectional models that satisfy conditional moment restrictions, and scalar diffusion processes.

No abstract

There are two difficulties with the implementation of the characteristic function-based estimators. First, the optimal instrument yielding the ML efficiency depends on the unknown probability density function. Second, the need to use a large set of moment conditions leads to the singularity of the covariance matrix. We resolve the two problems in the framework of GMM with a continuum of moment conditions. A new optimal instrument relies on the double indexing and, as a result, has a simple exponential form. The singularity problem is addressed via a penalization term. We introduce HAC-type estimators for non-Markov models. A simulated method of moments is proposed for nonanalytical cases. r

No abstract

Abstract:We propose a new estimator for the density of a random variable observed with an additive measurement error. This estimator is based on the spectral decomposition of the convolution operator, which is compact for an appropriate choice of reference spaces. The density is approximated by a sequence of orthonormal eigenfunctions of the convolution operator. The resulting estimator is shown to be consistent and asymptotically normal. While most estimation methods assume that the characteristic function (CF) of the error does not vanish, we relax this assumption and allow for isolated zeros. For instance, the CF of the uniform and the symmetrically truncated normal distributions have isolated zeros. We show that, in the presence of zeros, the problem is identi…ed even though the convolution operator is not one-to-one. We propose two consistent estimators of the density. We apply our method to the estimation of the measurement error density of hourly income collected from survey data.2

a b s t r a c tNonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models.We study this link using three measures of temporal dependence: ρ-mixing, β-mixing and α-mixing. Stationary diffusions that are ρ-mixing have mixing coefficients that decay exponentially to zero. When they fail to be ρ-mixing, they are still β-mixing and α-mixing; but coefficient decay is slower than exponential. For such processes we find transformations of the Markov states that have finite variances but infinite spectral densities at frequency zero. The resulting spectral densities behave like those of stochastic processes with long memory. Finally we show how state dependent, Poisson sampling alters the temporal dependence.