Mathematical Derivation of Significance Test of Unitary Linear Regression Equation Based on OLS
China-Asean Institute of Statistics, Guangxi University of Finance and Economics, Nanning 530003, China
| Abstract: | 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. |
| Keywords: | Unitary Linear Regression; OLS Estimator; Three Major Distributions; Test of Significance; Goodness of Fit |
| DOI: | 10.57237/j.wjms.2025.01.002 |
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Received: 19 January 2025
Accepted: 20 May 2025
Published Online: 30 June 2025
Funding
| 1. | 2022年度全国统计科学研究项目 (2022LY065) |
| 2. | 广西财经学院博士启动项目 (BS2021012) |
| 3. | 2024年度广西高等教育本科教学改革工程项目 (2024JGA324) |
| 4. | 广西财经学院校级本科教学改革工程项目 (2024XJJGYB16、2024XJJGYB22) |
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