Higher order asymptotics for the Central Limit Theorem and Large Deviation Principles

First, we present results that extend the classical theory of Edgeworth expansions to independent identically distributed non-lattice discrete random variables. We consider sums of independent identically distributed random variables whose distributions have (d+1) atoms and show that such distributions never admit an Edgeworth expansion of order d but for almost all parameters the Edgeworth expansion of order (d-1) is valid and the error of the order (d-1) Edgeworth expansion is typically O(n^{-d/2}) but the O(n^{-d/2}) terms have wild oscillations. Next, going a step further, we introduce a general theory of Edgeworth expansions for weakly dependent random variables. This gives us higher order asymptotics for the Central Limit Theorem for strongly ergodic Markov chains and for piece-wise expanding maps. In addition, alternative versions of asymptotic expansions are introduced in order to estimate errors when the classical expansions fail to hold. As applications, we obtain Local Limit Theorems and a Moderate Deviation Principle. Finally, we introduce asymptotic expansions for large deviations. For sufficiently regular weakly dependent random variables, we obtain higher order asymptotics (similar to Edgeworth Expansions) for Large Deviation Principles. In particular, we obtain asymptotic expansions for Cramer's classical Large Deviation Principle for independent identically distributed random variables, and for the Large Deviation Principle for strongly ergodic Markov chains

Sequential Data-Adaptive Bandwidth Selection by Cross-Validation for Nonparametric Prediction

We consider the problem of bandwidth selection by cross-validation from a sequential point of view in a nonparametric regression model. Having in mind that in applications one often aims at estimation, prediction and change detection simultaneously, we investigate that approach for sequential kernel smoothers in order to base these tasks on a single statistic. We provide uniform weak laws of large numbers and weak consistency results for the cross-validated bandwidth. Extensions to weakly dependent error terms are discussed as well. The errors may be {\alpha}-mixing or L2-near epoch dependent, which guarantees that the uniform convergence of the cross validation sum and the consistency of the cross-validated bandwidth hold true for a large class of time series. The method is illustrated by analyzing photovoltaic data.Comment: 26 page

Limit Theory for Moderate Deviations from a Unit Root under Weak Dependence

An asymptotic theory is given for autoregressive time series with weakly dependent innovations and a root of the form rho_{n} = 1+c/n^{alpha}, involving moderate deviations from unity when alpha in (0,1) and c in R are constant parameters. The limit theory combines a functional law to a diffusion on D[0,infinity) and a central limit theorem. For c > 0, the limit theory of the first order serial correlation coefficient is Cauchy and is invariant to both the distribution and the dependence structure of the innovations. To our knowledge, this is the first invariance principle of its kind for explosive processes. The rate of convergence is found to be n^{alpha}rho_{n}^{n}, which bridges asymptotic rate results for conventional local to unity cases (n) and explosive autoregressions ((1 + c)^{n}). For cCentral limit theory; Diffusion; Explosive autoregression, Local to unity; Moderate deviations, Unit root distribution, Weak dependence

A Bootstrap Theory for Weakly Integrated Processes

This paper develops a bootstrap theory for models including autoregressive time series with roots approaching to unity as the sample size increases. In particular, we consider the processes with roots converging to unity with rates slower than n?1. We call such processes weakly integrated processes. It is established that the bootstrap relying on the estimated autoregressive model is generally consistent for the weakly integrated processes. Both the sample and bootstrap statistics of the weakly integrated processes are shown to yield the same normal asymptotics. Moreover, for the asymptotically pivotal statistics of the weakly integrated processes, the bootstrap is expected to provide an asymptotic refinement and give better approximations for the finite sample distributions than the first order asymptotic theory. For the weakly integrated processes, the magnitudes of potential refinements by the bootstrap are shown to be proportional to the rate at which the root of the underlying process converges to unity. The order of boostrap refinement can be as large as o(n-1/2+_) for any espial > 0. Our theory helps to explain the actual improvements observed by many practitioners, which are made by the use of the bootstrap in analyzing the models with roots close to unity.

Robust Model Selection in Dynamic Models with an Application to Comparing Predictive Accuracy

A model selection procedure based on a general criterion function, with an example of the Kullback-Leibler Information Criterion (KLIC) using quasi-likelihood functions, is considered for dynamic non-nested models. We propose a robust test which generalizes Lien and Vuong's (1987) test with a Heteroscadasticity/Autocorrelation Consistent (HAC) variance estimator. We use the fixed-b asymptotics developed in Kiefer and Vogelsang (2005) to improve the asymptotic approximation to the sampling distribution of the test statistic. The fixed-b approach is compared with a bootstrap method and the standard normal approximation in Monte Carlo simulations. The fixed-b asymptotics and the bootstrap method are found to be markedly superior to the standard normal approximation. An empirical application for foreign exchange rate forecasting models is presented.

GMM for panel count data models

This paper gives an account of the recent literature on estimating models for panel count data. Specifically, the treatment of unobserved individual heterogeneity that is correlated with the explanatory variables and the presence of explanatory variables that are not strictly exogenous are central. Moment conditions are discussed for these type of problems that enable estimation of the parameters by GMM. As standard Wald tests based on efficient two-step GMM estimation results are known to have poor finite sample behaviour, alternative test procedures that have recently been proposed in the literature are evaluated by means of a Monte Carlo study.GMM, exponential models, hypothesis testing