Regression model: 𝑦 = 𝑎+ 𝑏𝑥+ 𝜖. Suppose that 𝑥 and 𝜖 are random

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Regression model: 𝑦 = 𝑎+ 𝑏𝑥+ 𝜖.
Suppose that 𝑥 and 𝜖 are random variables, and parameters 𝑎 and
𝑏 are constants.
1. If 𝑥 and 𝜖 are negatively correlated i.e., 𝐶𝑜𝑣(𝑥,𝜖) < 0,
how does it impact the estimation of parameter 𝑏 in the linear
regression model?
2. Suppose that the i.i.d samples (𝑥1,…,𝑥𝑛) are
realizations of the random variable 𝑥, and suppose that 𝑥 and 𝜖 are
independent. Given that
𝑉𝑎𝑟(𝑥)=𝐸[𝑥−𝐸(𝑥)]2= 1𝑛Σ(𝑥𝑖−𝑥̅)2𝑛𝑖=1
𝐶𝑜𝑣(𝑥,𝑦)= 𝐸[(𝑥−𝐸(𝑥))(𝑦−𝐸(𝑦))]=1𝑛Σ(𝑥𝑖−𝑥̅)(𝑦𝑖−𝑦̅)𝑛𝑖=1,
consider the expression for the Least Square Estimator 𝑏 in Page
42 of Lecture 8’s slides “Inventory Management III and
Forecasting”. Try to establish the expression for the estimator 𝑏
from the hints given above.