COLUMBIA UNIVERSITY STAT GR5205 FINAL EXAM

Document Content and Description Below

STAT GR5205 Final Exam Name: UNI: Please write your name and UNI The Fall GR5205 final is closed notes and closed book. Calculators are allowed. Tablets, phones, computers and other equivalent fo... rms of technology are strictly prohibited. Students are not allowed to communicate with anyone with the exception of the TA and the professor. If students violate these guidelines, they will receive a zero on this exam and potentially face more severe consequences. Students must include all relevant work in the handwritten problems to receive full credit. Theory Component Problem 1 [10 pts] Part I (5 pts) Let X be a full rank n × p design matrix and define the hat matrix as H = X(XTX)−1XT. Recall that the column space of X, denoted C(X), is the set of all linear combinations of the columns in X. Prove that if v 2 C(X), then Hv = v. 1Part II (5 pts) Let β^ be the least squares estimator of β. Use the result from Problem 1.I to prove that the sum of sample residuals is always zero, i.e., show Pn i=1 ei = 0. Problem 2 [25 pts] Consider three models: (1) Yi = β1xi1 + i; i = 1; : : : ; n; i iid ∼ N(0; σ2); (2) Yi = β2xi2 + i; i = 1; : : : ; n; i iid ∼ N(0; σ2); (3) Yi = β1xi1 + β2xi2 + i; i = 1; : : : ; n; i iid ∼ N(0; σ2): Denote the respective data vectors and full design matrix by Y = Y1 Y2 · · · Yn
T ; x1 = x11 x21 · · · xn1
T ; x2 = x12 x22 · · · xn2
T ; X = x1 x2
: Further, let H1, H2, and H be the respective hat-matrices of models (1), (2) and (3). 2Part I (10 pts) Assuming that the vectors x1 and x2 are perfectly uncorrelated (orthogonal), prove that HX = (H1 + H2) [Show More]

Last updated: 1 year ago

Preview 1 out of 19 pages