(Monte Carlo) Repeat the Monte Carlo experiment in Exercise 6.1 with the following changes.

(a) In addition to sample means, compute the sample variances and covariances of the three OLS fitted coefficients. Compare these values with their population counterparts.

(b) Also compute the average value of $s^2$ for each OLS fit and check whether this estimator appears to be unbiased for the conditional variance of $y_x$ given $x_i$.

(b) Also compute the average value of $2 for each OLS fit and check whether this estimator appears to be unbiased for the conditional variance of y, given.x.. 8.2 (Monte Carlo) Repeat the Monte Carlo experiment in Exercise 6.1, making each of the following adjustments separately. (a) Instead of ya, fit y+x, to the explanatory variables 1, x, and x. (b) Instead of y, fit y+x, to the explanatory variables I and x. (c) Generate x, from a uniform distribution on the interval (9. 10], instead of [1, 10]. Compare the sample means and variances of the fitted coefficients with those from the Monte Carlo experiment in Exercise 8.1. Try to explain your findings. 8.3 (Exact Multicollinearity) Consider a situation in which Assumptions 6.1 (First Moments, p. 110) and 7.1 (Second Moments, p. 130) hold but Assumption 3.1 (Full Rank, p. 53) is violated so that X is rank deficient. (a) What can we infer about the variance matrix of under such conditions? (b) How docs exact multicollinearity affect Var[|X]? (c) Show that El(-) (po)/rank(X) | X] = 0. Find an unbiased estimator for o allowing for exact multicollinearity. 8.4 Explain why Proposition 6 (Estimation of the Variance, p. 158) requires Assumption 6.1 whereas Proposition 5 (Variances of OIS, p. 157) does not. 8.5 (OLS Fitted Residuals) Show that Varly AX] [1-XX(XX)x] Also prove that this variance is less than or equal too. What transformation of the OLS fitted residuals would result in constant variances equal to o