We try to build a deeper understanding of the endogeneity issue in the regression model y...

 

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We try to build a deeper understanding of the endogeneity issue in the regression model y = a + bx + €. Suppose that x and € are random variables, and parameters a and b are constants. Cov(x,y)-Cov(x,E) (a). Show that b Var (x) = Hint: the covariance of the random variables Cov(x, y) has the following properties Cov(x, y) = Cov(x, a + bx + €) = Cov(x, a) + Cov(x, bx) + Cov(x, e), Cov(x, bx)=bVar(x). (b). If x and e are independent, what is Cov(x, e)? (c). If x and are negatively correlated i.e., Cov(x, e) < 0, how does it impact the estimation of parameter b in the linear regression model? (d). Suppose that the i.i.d samples (x₁,...,xn) are realizations of the random variable x, and suppose that x and e are independent. Given that Var(x) = E[x - E(x)]² = ₁₁(x₁ - x)² i=1 Cov(x, y) = E[(x - E(x))(y-E(y))] = Σ₁₁(x₁ - x)(y₁ - y), consider the expression for the Least Square Estimator b in Page 31 of Lecture 8's slides "Forecasting". Try to establish the expression for the estimator b from the hints given above. We try to build a deeper understanding of the endogeneity issue in the regression model y = a + bx + €. Suppose that x and € are random variables, and parameters a and b are constants. Cov(x,y)-Cov(x,E) (a). Show that b Var (x) = Hint: the covariance of the random variables Cov(x, y) has the following properties Cov(x, y) = Cov(x, a + bx + €) = Cov(x, a) + Cov(x, bx) + Cov(x, e), Cov(x, bx)=bVar(x). (b). If x and e are independent, what is Cov(x, e)? (c). If x and are negatively correlated i.e., Cov(x, e) < 0, how does it impact the estimation of parameter b in the linear regression model? (d). Suppose that the i.i.d samples (x₁,...,xn) are realizations of the random variable x, and suppose that x and e are independent. Given that Var(x) = E[x - E(x)]² = ₁₁(x₁ - x)² i=1 Cov(x, y) = E[(x - E(x))(y-E(y))] = Σ₁₁(x₁ - x)(y₁ - y), consider the expression for the Least Square Estimator b in Page 31 of Lecture 8's slides "Forecasting". Try to establish the expression for the estimator b from the hints given above.

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a To show that b Covxy Covxe Varx well use the properties of covariance mentioned in the hint We sta... View the full answer