1. Review: Assumptions under linear regression

1.1. Stat’s version

Source: https://online.stat.psu.edu/stat501/Lesson07#mlr-model-assumptions

• 第一条(linear function)包含了两点：1）linear relationship，2）zero conditional mean
• 四条合起来也是：1）linear relationship，2）x⊥e，3）e iid normal(0,σ2)

1.2. Wooldridge’s version (Econometrics)

Source: Introduction Econometrics by Jeffrey M. Wooldridge

• MLR1 ~ MLR4：Unbiasedness of OLS
• MLR1 ~ MLR5：OLS is BLUE (best linear unbiased estimator)
  • 统称为Gauss-Markov assumptions
  • 引入MLR5的两个作用：1）简化variance的表达式；2）保证OLS的efficiency
• MLR1 ~ MLR6：OLS is MVUE (minimum variance unbiased estimator)
  • 统称为Classical linear model assumptions
  • MLR6非常强，包含了MLR4和MLR5；但习惯上还是将4和5保留
  • 即使MLR6不成立（i.e., 只有MLR1~MLR5），在large sample size下，OLS is approimately normal

2. Visualization

2.1. Residuals vs. fitted values