Mathematical Derivation of Significance Test of Unitary Linear Regression Equation Based on OLS

Unitary linear regression plays a unique and important position in the teaching content system of statistics and econometrics. It is an important basis for understanding and mastering multiple linear regression. However, as the key and difficult point of unitary linear regression, the properties of OLS estimators and the significance test of regression equations often lack of systematic mathematical derivation. In order to derive systematically the significance test of regression equation, the paper first combs the five basic assumptions of Gauss and the definition and mathematical characteristics of three major distributions (Chi-square distribution, t-distribution and F-distribution). Then, it is demonstrated that the OLS estimator of the regression equation satisfies the characteristics of linearity, unbias and effectiveness, and it is the optimal linear unbiased estimator (BLUE). Then, expounds the determination coefficient method and the estimation standard error method about the judgement on goodness of fit for regression line. The two methods have intrinsic consistency. The smaller the estimation standard error is, the larger the determination coefficient is, which means the better the fitting degree of regression line. Finally, by proving that SSR/σ2 and SSE/σ2 obey the Chi-square distribution of freedom of 1 and (n-2) respectively, it is proved that t statistic and F statistic obey the t distribution and F distribution respectively, and the F test is consistent with the t test and the significance test of correlation coefficient r. Therefore, in the statistical testing of unitary linear regression equations, it is not necessary to test the statistical significance of the correlation coefficient, regression coefficient, and overall regression equation simultaneously. Only one of the three needs to be tested. 一元线性回归在统计学和计量经济学的教学内容体系中具有独特的重要地位，它是理解和掌握多元线性回归的重要基础。然而，作为一元线性回归的重点和难点，关于OLS估计量性质与回归方程显著性检验往往缺乏系统的数学推导。为了对回归方程显著性检验进行系统的数学推导，论文首先梳理了高斯的五个基本假定和三大分布（卡方分布、t分布和F分布）的定义与数学特征；接着论证了回归方程OLS估计量满足线性特性、无偏性和有效性的特点，是最优线性无偏估计量（BLUE）；然后阐述了关于回归直线拟合优度判断的判定系数法和估计标准误差法，两种方法具有内在的一致性，估计标准误差越小，判定系数随之越大，意味着回归直线的拟合程度越好；最后通过证明SSR/σ2和SSE/σ2分别服从自由度为1和(n-2)的卡方分布，由此证明了t统计量和F统计量分别服从t分布和F分布，而且F检验与t检验、相关系数r显著性检验具有一致性。因此，在一元线性回归方程的统计检验中，不需要对整体线性关系、回归系数和相关系数同时进行统计显著性检验，只需要对三者之一进行统计检验即可。