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以分数过程为积分器的随机积分表示与弱收敛AI 解读

This paper considers the asymptotic distribution of the sample covariance of a nonstationary fractionally integrated process with the stationary increments of another such process—possibly itself. Questions of interest include the relationship between the harmonic representation of these random variables, which we have analyzed in a previous paper (Davidson and Hashimzade, 2008), and the construction derived from moving average representations in the time domain. Depending…

Breitung 统计量极限分布的一般界AI 解读

We consider the Breitung (2002, Journal of Econometrics 108, 343–363) statistic ξ n , which provides a nonparametric test of the I(1) hypothesis. If ξ denotes the limit in distribution of ξ n as n → ∞, we prove (Theorem 1) that 0 ≤ ξ ≤ 1/π 2 , a result that holds under any assumption on the underlying random variables. The result is a special case of a more general result (Theorem 3), which we prove using the so-called cotangent method associated with Cauchy's residue…

分数布朗运动的不同频域与时域版本AI 解读

This paper compares models of fractional processes and associated weak convergence results based on moving average representations in the time domain with spectral representations. Both approaches have been applied in the literature on fractional processes. We point out that the conventional forms of these models are not equivalent, as is commonly assumed, even under a Gaussianity assumption. We show that it is necessary to distinguish between “two-sided” processes depending…

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线性条件异方差模型的矩与记忆性质及一种新模型AI 解读

conditional heteroscedasticity models in which the conditional variance is a linear function of squared lags of the process. GARCH, IGARCH, FIGARCH, and a newly proposed generalization, the HYGARCH model, belong to this class. Conditions are derived for the existence of second and fourth moments, and for the limited memory condition of near-epoch dependence. The HYGARCH model is applied to 10 daily dollar exchange rates, and also to data for Asian exchange rates over the…

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半参数线性过程之和的核方差估计量的一致性AI 解读

Conditions are derived for the consistency of kernel estimators of the variance of a sum of dependent heterogeneous random variables, with a representation as moving averages of near-epoch dependent functions of a mixing process. Fourth moments are not generally required. The conditions permit more dependence than a purely non-parametric representation allows, and may be close to those of the best-known conditions for the functional central limit theorem. The class of…

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函数中心极限定理与随机积分的弱收敛（二）AI 解读

This paper derives a functional central limit theorem for the partial sums of fractionally integrated processes, otherwise known as I ( d ) processes for | d | < 1/2. Such processes have long memory, and the limit distribution is the so-called fractional Brownian motion, having correlated increments even asymptotically. The underlying shock variables may themselves exhibit quite general weak dependence by being near-epoch-dependent functions of mixing processes. Several weak…

函数型中心极限定理与随机积分的弱收敛（一）AI 解读

This paper gives new conditions for the functional central limit theorem, and weak convergence of stochastic integrals, for near-epoch-dependent functions of mixing processes. These results have fundamental applications in the theory of unit root testing and cointegrating regressions. The conditions given improve on existing results in the literature in terms of the amount of dependence and heterogeneity permitted, and in particular, these appear to be the first such…

异方差与自相关协方差矩阵核估计量的一致性AI 解读

Conditions are derived for the consistency of kernel estimators of the covariance matrix of a sum of vectors of dependent heterogeneous random variables, which match those of the currently best-known conditions for the central limit theorem, as required for a unified theory of asymptotic inference. These include finite moments of order no more than 2 + for > 0, trending variances, and variables which are near-epoch dependent on a mixing process, but not necessarily mixing.…

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混合过程近历相依函数的全局非平稳中心极限定理：渐近退化情形AI 解读

The central limit theorem in Davidson [2] is extended to allow cases where the variances of sequence coordinates can be tending to zero. A trade-off is demonstrated between the degree of dependence and the rate of degeneration. For the martingale difference case, it is sufficient for the sum of the variances to diverge at the rate of log n .

混合过程的全局非平稳近历元相依函数的中心极限定理AI 解读

A central limit theorem is proved for dependent stochastic processes. Global heterogeneity of the distribution of the terms is permitted, including asymptotically unbounded moments. The approach is to adapt a CLT for martingale differences due to McLeish and show that suitably defined Bernstein blocks satisfy the required conditions.