Asymptotic Equivalence of Ordinary Least Squares and Generalized Least Squares in Regressions with Integrated Regressors
含单整回归变量时普通最小二乘与广义最小二乘的渐近等价性AI 解读 ↓
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含单整回归变量时普通最小二乘与广义最小二乘的渐近等价性
- 研究问题
- 当回归变量为多维单整过程、误差具有平稳自回归相关性时，OLS与GLS是否具有相同极限分布，常规显著性检验如何调整？
- 方法与思路
- 作者利用多元不变性原理，以向量布朗运动的泛函表示估计量的渐近分布，并讨论误差长期方差在假设检验中的作用。
- 材料陈述的贡献
- 摘要称，OLS与GLS的渐近分布相同，并将既有双变量结果推广至多元回归；以一致的长期方差估计替换通常方差估计可修正相关检验。
- 阅读关注点
- 应核对回归变量与误差的独立性条件、不同标准化方式及长期方差估计要求；摘要中部分估计量和泛函公式缺失，须查原PDF。
- 阅读边界
- 仅依据稿件版本未注明的OpenAlex索引摘要。公式存在缺失，不能重构标准化或极限分布，也不能将渐近等价解释为有限样本表现相同。
原文摘要
It is shown that in the multiple regression model y 1 = x 1′β + u t, where u t, is a stationary autoregressive process and x 1 is an integrated m-vector process, the asymptotic distributions of the ordinary least squares (OLS) and generalized least squares (GLS) estimators of β are identical. This generalizes a result obtained by Krämer (1986) for two-variate regression and extends fixed regressor theory developed by Grenander and Rosenblatt (1957). Our approach uses a multivariate invariance principle and yields explicit representations of the asymptotic distributions in terms of functionals of vector Brownian motion. We also provide some useful asymptotic results for hypothesis tests of the model. Thus if x 1 is generated by a vector (autoregressive integrated moving average) ARIMA(r, l, s) model and u t is generated by an independent (autoregressive) AR(p) process, then and have the same limiting distribution (where and are the OLS and GLS estimators, respectively). This distribution is nonnormal and most conveniently represented in terms of a vector Brownian motion (B 1(r), B 2(r)′) as the functional , where B 2 is an m-vector Brownian motion independent of B 1. Furthermore, if X(T × m) is a matrix of T observations of x 1 then and have the same limiting normal distribution as T ↑ ∞. But the variance of this normal distribution is given by σ12 = 2πfu(0) [where f u(λ) is the spectral density of u t] and not σ2 = var(u t). Traditionally constructed asymptotic tests of significance are invalid in the present context; however, these tests may be made robust by simply replacing the usual estimators of σ2 with consistent estimates of σ12 .