[試題] 100上 陳釗而 數量方法入門 期中考
課程名稱︰數量方法入門
課程性質︰必修
課程教師︰陳釗而
開課學院:社會科學院
開課系所︰經濟所
考試日期(年月日)︰100.09.02
考試時限(分鐘):90
是否需發放獎勵金:是
(如未明確表示,則不予發放)
試題 :
1. For any x,y∈R^n, show that ║x+y║<= ║x║+║y║.
2. Briefly explain the Projection Theorem (in mathematics or in words).
3. Let X be a matrix of order n*k and rank k.
(a)Verify that M = I-Px is an orthogonal projection matrix,
-1
where Px = X(X'X)X'.
(b)Verify that MX = 0.
(c)Rank(M) = ?
4. (Eigenvalues, idempotent matrices, and symmetry)
(a)Show that the eigenvalue of an idempotent matrix are 0 and 1.
(b)Show that if A is symmetric and has only eigenvalues 0 and 1,
it is idempotent.
5. Show that if a matrix P is symmetric and idempotent, then P is
positive semi-definite.
-1
6. Let A be a nonsingular matrix. Consider the partition of A and matrix A
-1 [ -1 -1 -1 ]
A = [ W (-W)(A11)(A22) ]
[ -1 -1 -1 -1 -1 -1 ]
[ (-A22)(A21)W (A22)+(A22)(A21)(W)(A12)(A22) ]
where A = [ A11 A12 ]
[ A21 A22 ]
is a partition of A such that A11 and A22 are square matrices, and we define
-1
W = A11-(A12)(A22)(A21). Verify that for the linear regression model
y = X1*β1 + X2β2 + ε
the least squares estimator for β1 is given by
^ -1 -1
β1 = (X1'(I-Px2)X1)X1'(I-Px2)y, where Px2 = X2(X2'X2)X2'.
^ ^ ^ -1
Hint: X = [X1 X2], [β1 β2]' = β = (X'X)X'y,
where X is a matrix of order n*k, X1 is a matrix of order n*k1,
X2 is a matrix of order n*k2, β is an k*1 vector, β1 is an k1*1 vector,
β2 is an k2*1 vector, and k1+k2 = k.
7. Let X be a matrix of order n*k and rank k, and let V be symmetric and
positive difinite of order n.
(a)Show that the matrix
-1-1
Q:= I-V^(-1/2)X(X'VX)X'V^(-1/2)
is symmetric and idempotent.
(b)Use part (a) and the fact that X'V^(1/2)QV^(1/2)X = 0 to show that
-1-1 -1 -1
(X'VX)X'V = (X'X)X'
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