Problem Set 3
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Problem Set 3
1. Consider the model
y = Xg + ee
(a) Using the Method of Moments, derive the estimator of g , gˆuu . (b) Using the OLS method, derive the estimator of g , gˆoLs .
(c) Discuss the relationship between the two methods.
2. In the OLS method, discuss why (X1X) is invertible.
3. Let z1 = (2. 1. 3) and y1 = (2. 3. 5). Let the orthogonal projection of y on the span of z be P .
(a) Find where Py = z.
(b) Find P .
(c) Find the projection ○ that is orthogonal to the projection P .
4. Consider two regressions
y = wa + ε
y = Xg + ε
where R(w) = R(X), or equivalently, w = XT for some nonsingular matrix T.
(a) Show that the two regressions have the same fitted values (both yˆ and εˆ), and R2 .
(b) Show that = T_1gˆ. Interpret the result.
5. Consider the regresion
y = uu + Xg + e
(a) Show that the OLS estimator gˆ of g can be written as gˆ = (X*/ X* )_1X*/ y* where the i-th row of y* and X* are, respectively, given by
yi(*) = yi _ y¯ and Xi(*)/ = Xi(1) _ 1
(b) Show that the OLS estimator of u is
= y¯ _ 1gˆ
where is a k x 1 sample mean vector.
2022-06-10