6.4 The Sampling, or Probability, Distributions of OLS Estimators 1. One more assumption of the CLRM needed A6.5. In the PRF Y=B+B2X+Wi the error term Hi follows the normal distribution with mean zero and variance 2. That is H N(O, 2)(6.17) Central limit theorem If there is a large number of independent and identical distributed random variables, then, with a few exceptions, the distribution of their sum tends to be a normal distribution as the number of such variables increases indefinitely 2. b, and b, follow normal distribution u follows the distribution --b, and b, are linear functions of the normally distributed variable H b, and b, are normally distributed bNB c07 (6.18) 2=var( (1)=x (6.4) b,N B,,O (6.19) 02=var(b2) (66) b X6.4 The Sampling , or Probability, Distributions of OLS Estimators ◼ 1.One more assumption of the CLRM.needed: A6.5. In the PRF Yi=B1+B2Xi+μi, the error term μi follows the normal distributionwith mean zero and variance . That is μi~N(0, ) (6.17) ◼ Central limit theorem: ——If there is a large number of independent and identically distributed random variables, then, with a few exceptions, the distribution of their sum tends to be a normal distribution as the number of such variables increases indefinitely. ◼ 2. b1 and b2 follownormal distribution ∵---μ followsthe distribution ---b1 and b2 are linear functions of the normally distributed variable μ, ∴b1 and b2 are normally distributed. b1~N(B1 , ) (6.18) =var(b1 )= (6.4) b2~N(B2 , ) (6.19) =var(b2 )= (6.6) 2 σ σ 2 2 b1 σ 2 b1 σ   2 i 2 i n χ X 2 b2 σ 2 b2 σ  2 i 2 χ σ