Prior Information on the Coefficients when the Disturbance Covariance Matrix is Unknown
扰动协方差矩阵未知时的系数先验信息AI 解读 ↓
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扰动协方差矩阵未知时的系数先验信息
- 研究问题
- 在线性回归扰动协方差结构未知或被误设时，即使施加正确的系数线性约束，是否仍可能降低某些参数函数的估计效率？
- 方法与思路
- 作者比较先验约束不同的最小二乘估计，考察线性模型确定性部分与扰动协方差设定的关系，并用正则矩阵束理论推导结果。
- 材料陈述的贡献
- 摘要称，当扰动协方差实质偏离高斯–马尔可夫条件时，正确的系数线性约束也可能使某些可估参数函数的估计效率降低。
- 阅读关注点
- 应核对效率比较采用的准则、受约束与未受约束估计量、协方差误设的具体条件，以及哪些可估参数函数会受到影响。
- 阅读边界
- 摘要未列出完整定理条件或应用细节；结论针对特定协方差误设下的效率比较，不能据此断言所有先验信息或贝叶斯方法都有害。
原文摘要
usefulness of prior information. Viewed broadly, this theme has encompassed such diverse topics as the estimation of a system of simultaneous structural equations, the specification and estimation of distributed lags, and the formal integration of stochastic prior and sample information through Bayes' theorem. Without exception, the results have encouraged the incorporation of further prior information into our statistical procedures, in the sense that the judicious use of such information has produced unambiguously better estimators. That this need not be the case in a typical linear regression application is thus somewhat surprising and constitutes the topic of this paper. Here, we develop the relationship between the specification of the deterministic and the stochastic components of a linear model and show that whenever the covariance structure of the disturbance process is effectively misspecified,2 one can no longer justify the use of prior information about the deterministic part of the model. If the error covariance matrix differs substantively from that required by the Gauss-Markov theorem, the imposition of correct linear restrictions on the regression coefficients leads to less efficient estimators of some estimable functions of the parameters.3 Prior information can hurt! We begin by introducing notation and examining the efficiency of least squares estimators in linear models with varying amounts of prior information. An application of the theory of regular pencils then produces the main results (Section 3), practical implications are drawn in Section 4, and we conclude with an application (Section 5).