Econometrics 710 midterm exam 2000 take the linear

Question

Econometrics 710 Midterm Exam 2000

1. The model is

y_i = xi_β + e_i           E (e_i| x_i ) = 0

where x_i , β and e_i are scalar. We consider the estimator

β^˜ = y^-/x = _i=1Σⁿy_i/_i=1Σⁿx_i

We assume that x_i and e_i have finite fourth moments and that {y_i, x_i} are a random sample (iid).

(a) Find E(β^˜| X).

(b) Find V ar (β^˜| X).

(c)  Show that β^˜ →_p β as n → ∞. Does this require any additional assumptions?

(d) Find the asymptotic distribution of √n(β^˜ - β). as n →∞.

(e) Without imposing any additional assumptions, is β^˜ necessarily less efficient than OLS? (By efficiency, I mean lower asymptotic variance.)

2. Take the linear regression Y = Xβ + e with E(e_i | x_i) = 0. Let θ = 1/β₁where β₁ is the first element of β. Let β^ˆ be the OLS estimator of β and V^ˆ be the estimator of V ar (β)^ˆ. Find au asymptotically valid 95% confidence interval for θ. (Give the explicit formula as a function of β^ˆ aud V^ˆ.)

3. In the linear regression Y = Xβ + e with E(e_i|x_i) = 0, it is known that the true β satisfies the restriction

Rβ = 0

where R is a q × k matrix with q < k. Consider the estimator

β^˜ = β^ˆ - (X'X)^-1 R'[R (X'X)^-1R']^-1Rβˆ.

(a) Show that Rβ^˜ = 0.

(b) Find E(β^˜|X).

(s) Find V ar (β^˜| X). [Hint: First write β^˜ as a linear function of βˆ.]

(d) Give an expression for a valid standard error for the elements of β^˜. You do not need to give a proof of validity.

4. Take the linear regression Y = Xβ + e with E(e_i|x_i) = 0. For one particular value of x, the object of interest is the conditional mean

E(y_i|x_i = x) = g(x).

Describe how you would use the percentile-t bootstrap to construct a confidence interval for g(x).