3 Large sample properties of the lse Consider the linear regression model M=t+Et,Et~id(0,2),t=1,……,m (a) Show that the least squares estimat or of B, b is consistent (b) Derive the asy mptot ic distribution of b 2. For the linear regression model d an estimator is considered. Assume Xt) is a sequence of const ants. (a) What assumptions are required for the consist ency of b (b) Derive the asymptot ic distribution of b 3. Suppose that Xi have a binomial distribut ion b(m, p) and that X1 ndependent (a)What is the probability limit of X=I21X? b)What is the limiting distribution of vn(X-mp 4. Show that the mean square convergence implies the convergence in proba bility 5. Consider the linear regression model iid hen( Xi is a sequence of constants. What assumpt ions are required for the consistency of the LsE of 6? Are these assumptions reasonable for empirical analySIS 6. Consider the linear regression model Is the Ols estimat or of B consistent?4 3 Large sample properties of the LSE 1. Consider the linear regression model yt = βt + εt , εt ∼ iid
0, σ2 , t = 1, · · · , n. (a) Show that the least squares estimator of β, b is consistent. (b) Derive the asymptotic distribution of b. 2. For the linear regression model yt = βxt + εt , εt ∼ iid
0, σ2 , t = 1, · · · , n, an estimator ¯b = n t=1 yt n t=1 xt is considered. Assume {Xt} is a sequence of constants. (a) What assumptions are required for the consistency of ¯b. (b) Derive the asymptotic distribution of ¯b. 3. Suppose that Xi have a binomial distribution b (m, p) and that X1, · · · , Xn are independent. (a) What is the probability limit of X¯ = 1 n n i=1 Xi? (b) What is the limiting distribution of √ n
X¯ − mp ? 4. Show that the mean square convergence implies the convergence in probability. 5. Consider the linear regression model yi = β xi + εi , εi ∼ iid
0, σ2 , i = 1, · · · , n when {Xi} is a sequence of constants. What assumptions are required for the consistency of the LSE of β? Are these assumptions reasonable for empirical analysis? 6. Consider the linear regression model yt = β t + εt , εt ∼ iid
0, σ2 , t = 1, · · · , n. Is the OLS estimator of β consistent?