hw2 - Winter 2010 STAT 331/SYDE 334 Zhu M University of...

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Winter 2010 STAT 331/SYDE 334 Zhu, M University of Waterloo STAT 331/SYDE 334 Problem Set 2 Due: F, Feb 12, 2010 or Tu, Feb 16, 2010 Hand in this assignment into a box labelled “STAT 331” in MC 6028. If you are leaving Waterloo for the reading week, please do so on F, Feb 12 by 3:00 pm. If you are not leaving Waterloo for the reading week, you can do so on Tu, Feb 16 by 12:00 noon. All data sets are available in a directory called “Data Sets” on the course web site hosted by the ANGEL system, http://uwace.uwaterloo.ca/ . Once logged onto the system, click on the “Content” tab. 1. (12 pts) Consider the linear model y = X β + ǫ under the Gauss-Markov condition, i.e., E ( ǫ ) = 0 and Var( ǫ ) = σ 2 I . (a) (8 pts) Let ˆ β = ( X T X ) - 1 X T y be the usual least-squares estimate. Use the formula Cov( Ay , By ) = A Var( y ) B T to show Cov( y ˆ y , ˆ β ) = O , where O is a matrix of all zeros. (b) (2 pts) What extra condition, if any, is needed if you want to conclude that y ˆ y and ˆ β are statistically independent? (c) (2 pts) In the general context of the study of linear models, briefly comment why it is sometimes important for y ˆ y and ˆ β to be statistically independent.
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