On the Asymptotic Efficiency of GMM

This paper derives conditions under which the generalized method of moments (GMM) estimator is as efficient as the maximum likelihood estimator (MLE). The data are supposed to be drawn from a parametric family and to be stationary Markov. We study the efficiency of GMM in a general framework where the set of moment conditions may be finite, countable infinite, or a continuum. Our main result is the following. GMM estimator is efficient if and only if the true score belongs to the closure of the linear space spanned by the moment conditions. This result extends former ones in two dimensions: (a) the moments may be correlated, (b) the number of moment restrictions may be infinite. It suggests a way to construct estimators that are as efficient as MLE. In the last part of this paper, we show how to calculate the greatest lower bound of instrumental variable estimatorsAsymptotic efficiency, GMM, infinity of moment conditions, reproducing kernel Hilbert space, efficiency bound.

Nonlinearity and Temporal Dependence

Nonlinearities in the drift and diffusion coefficients influence temporal dependence in scalar diffusion models. We study this link using two notions of temporal dependence: beta-mixing and rho-mixing. We show that beta-mixing and rho-mixing with exponential decay are essentially equivalent concepts for scalar diffusions. For stationary diffusions that fail to be rho-mixing, we show that they are still beta-mixing except that the decay rates are slower than exponential. For such processes we find transformations of the Markov states that have finite variances but infinite spectral densities at frequency zero. Some have spectral densities that diverge at frequency zero in a manner similar to that of stochastic processes with long memory. Finally we show how nonlinear, state-dependent, Poisson sampling alters the unconditional distribution as well as the temporal dependence.Mixing, Diffusion, Strong dependence, Long memory, Poisson sampling

Nonlinearity and Temporal Dependence

Nonlinearities in the drift and diffusion coefficients influence temporal dependence in scalar diffusion models. We study this link using two notions of temporal dependence: β−mixing and ρ−mixing. Weshow that β−mixing and ρ−mixing with exponential decay are essentially equivalent concepts for scalar diffusions. For stationary diffusions that fail to be ρ−mixing, we show that they are still β−mixing except that the decay rates are slower than exponential. For such processes we find transformations of the Markov states that have finite variances but infinite spectral densities at frequency zero. Some have spectral densities that diverge at frequency zero in a manner similar to that of stochastic processes with long memory. Finally we show how nonlinear, state-dependent, Poisson sampling alters the unconditional distribution as well as the temporal dependence. Les non-linéarités dans les coefficients de mouvement et de diffusion ont une incidence sur la dépendance temporelle dans le cas des modèles de diffusion scalaire. Nous examinons ce lien en recourant à deux notions de dépendance temporelle : mélange β et mélange ρ. Nous démontrons que le mélange β et le mélange ρ avec dégradation exponentielle constituent des concepts fondamentalement équivalents en ce qui a trait aux diffusions scalaires. Pour ce qui est des diffusions stationnaires qui ne se classent pas dans le mélange ρ, nous démontrons quâelles appartiennent quand même au mélange β, sauf que les taux de dégradation sont lents plutôt quâexponentiels. Pour ce genre de processus, nous recourons à des transformations des états de Markov dont les variations sont finies, mais dont les densités spectrales sont infinies à la fréquence zéro. Certains états ont des densités spectrales qui divergent à la fréquence zéro de la même façon que dans le cas des processus stochastiques à mémoire longue. En terminant, nous indiquons la façon dont lâéchantillonnage de Poisson qui est non linéaire et dépendant de lâétat modifie la distribution inconditionnelle et la dépendance temporelle.Mixing, Diffusion, Strong dependence, Long memory, Poisson sampling., mélange, diffusion, forte dépendance, mémoire longue, échantillonnage de Poisson.

Nonlinearity and Temporal Dependence

Nonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models. We study this link using three measures of temporal dependence: rho-mixing, beta-mixing and alpha-mixing. Stationary diffusions that are rho-mixing have mixing coefficients that decay exponentially to zero. When they fail to be rho-mixing, they are still beta-mixing and alpha-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.Diffusion, Strong dependence, Long memory, Poisson sampling, Quadratic forms

Efficient Estimation with Many Weak Instruments Using Regularization Techniques

The problem of weak instruments is due to a very small concentration parameter. To boost the concentration parameter, we propose to increase the number of instruments to a large number or even up to a continuum. However, in finite samples, the inclusion of an excessive number of moments may be harmful. To address this issue, we use regularization techniques as in Carrasco (2012) and Carrasco and Tchuente (2014). We show that normalized regularized two-stage least squares (2SLS) and limited maximum likelihood (LIML) are consistent and asymptotically normally distributed. Moreover, our estimators are asymptotically more efficient than most competing estimators. Our simulations show that the leading regularized estimators (LF and T of LIML) work very well (are nearly median unbiased) even in the case of relatively weak instruments. An application to the effect of institutions on output growth completes the article

Nonparametric estimation of the density of a change-point

The paper considers a panel model where the regression coe¢ cients undergo changes at an unknown time point, di§erentfor each series. The timings of changes are assumed to be independent, identically distributed, and drawn from some com-mon distribution, the density of which we aim to estimate nonparametrically. The estimation procedure involves two steps. First, changepoints are estimated indi-vidually for each series using the least-squares method. While these estimators are not consistent, they can be regarded as noisy signals of the true change-points. To address the inherent estimation error, a deconvolution kernel estimator is applied to estimate the density of the change-point. The paper establishes the consistency of this estimator and demonstrates that the rate of convergence of the Mean Inte-grated Squared error (MISE) is faster than that obtained with normal or Laplacian errors. Finally, using a Bayesian approach, we propose an estimator of the poste-rior means of the breakpoints, utilizing nonparametric estimates of the required densities. An application of the proposed methodology to portfolio returns reveals how quickly the markets responded to the Covid shock

On the Asymptotic Efficiency of GMM

This paper derives conditions under which the generalized method of moments (GMM) estimator is as efficient as the maximum likelihood esti-mator (MLE). The data are supposed to be drawn from a parametric family and to be stationary Markov. We study the efficiency of GMM in a general framework where the set of moment conditions may be finite, countable infinite, or a continuum. Our main result is the following. GMM estimator is efficient if and only if the true score belongs to the closure of the linear space spanned by the moment conditions. This result extends former ones in two dimensions: (a) the moments may be correlated, (b) the number of moment restrictions may be infinite. It suggests a way to construct estima-tors that are as efficient as MLE. In the last part of this paper, we show how to calculate the greatest lower bound of instrumental variable estimators