Determination of Linear Relations between Systematic Parts of Variables with Errors of Observation the Variances of Which Are Unknown
观测误差方差未知时变量系统部分间线性关系的确定AI 解读 ↓
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观测误差方差未知时变量系统部分间线性关系的确定
- 研究问题
- 当观测误差方差未知时，能否借助工具变量一致估计变量系统部分之间的线性关系，并评估这些估计在经济时间序列中的推断准确性？
- 方法与思路
- 作者利用足够数量的相关工具变量讨论概率收敛；在正态分布及特定假设下求两变量情形的精确分布，并比较时间序列模型、扩展至多变量。
- 材料陈述的贡献
- 摘要称，误差方差未知时仍可获得一致估计，并给出特定情形的有限样本分布、多变量扩展及非小样本检验，同时展示时间序列应用。
- 阅读关注点
- 读全文应核对工具变量有效性与数量条件、精确分布所需假设、时间依赖对推断的影响，以及样本长度、误差方差和工具相关性如何影响精度。
- 阅读边界
- 精确分布仅在摘要列明的两个研究变量、一个工具变量及正态条件等假设下成立；“序列不太短、误差不太大”未给出阈值或适用边界。
原文摘要
Given a sufficient number of instrumental variables significantly correlated with the\ninvestigational variables, consistent estimates of the coefficients of the linear relations can be determined (if they exist), without knowledge of the disturbance variances. The estimates are discussed from the viewpoint of probability convergence. In the case of two investigational and one instrumental variable, all three variables distributed on the normal surface, the distribution of the estimate of the coefficient is found exactly for all sample sizes, on certain hypotheses. The distribution function is remarkably simple. The applicability of the theorem to economic time series is discussed by (a) comparing the probability inferences derived from this Model A with those for the simplest stationary time-series model, termed Model B, and (b) by comparing the large-sample variances on several models. It is found that the theory can be used with confidence when the series are not too short and the error variances not too large. The theory is applied to a particular time series, showing that the accuracy of the estimate of the coefficient depends on the correlation between the instrumental variable and the two investigational variables. The theory to which reference is made in Sections II, III, and IV, relating to the two-investigational-variable case, is extended to many variables and tests are given, applicable when samples are not small, for determining the significance of coefficient estimates.