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Problem 1 (5+5+ 10+ 5+5+ 1 5+ 5: 50 points) Consider a multiple linear regression problem with intercept (i.e., the design matrix X E Rnxlp'l'l) has

Questions a-g is needed.

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Problem 1 (5+5+ 10+ 5+5+ 1 5+ 5: 50 points) Consider a multiple linear regression problem with intercept (i.e., the design matrix X E Rnxlp‘l‘l)
has a column of all ones, and p other columns corresponding to p covariates). Given the following
information: 2 0 U 0 —l
12_ 1T_010.5n 1T_—3 _
Elly&quot; —18.593. ;[X X]— I] 05 1 D , ”X y— 1 , n—lfi
0 U U 1 0.1 answer the following questions: [a] Is ﬁ£XTX:} PSD? Justify your answer. (b) 15 x all—rank? Justify your answer. (13) Find 5', the lemit~squares estimate (LSE) of :3. (d) Find the covariance matrix of 8.1:; this matrix PSD?
(e) Let us write 3 = [ﬁmﬁhﬁmiég]. What is mvuélﬁg)?
( f) Find a 95% conﬁdence interval for 32. Given this level. of conﬁdence should we accept or reject
the hypothesis: ﬁg 2 f}? (g) Let H be the orthogonal projection matrix, projecting onto lrn(X]. What is the dimension of
Im(lr — H J?

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