High Dimensional Data Analysis Individual Assignment: S2, 2024
Department of Econometrics and Business Statistics, Monash University
Due Date: 11th October 2024 at 4:30PM
Questions for ETF3500 and ETF5500 students
1. Consider the n× 3 data matrix Y The data matrix stores n observations of the 3 dimensional vector
y = (y1, y2, y3). The data matrix has been demeaned. Use this information to answer the following
questions. (5 marks).
• S denote the sample covariance matrix of Y. State the sample covariance matrix of Y. Be sure to
clearly define any quantities needed to state S, as well as the dimension of S.
• What is the dimension of an arbitrary eigenvector associated with S?
• Let X = Yβ, where β is a (3× 1) vector. What is the dimension of the sample covariance matrix of X?
State the sample covariance matrix of X in terms of S.
2. Let w be the eigenvector of S corresponding to the largest eigenvalue. In addition, let v be any
arbitrary column vector, whose dimension is the same as that of w. Discuss whether the following
matrix operations are conformable. If the operation is conformable, state the dimension of the product.
(5 marks)
• w′Y
• ww′
• wv′
• S′Y
• Yw+ v′
3. Let X = YC′, where
C =
[
w′
u′
]
,
w is defined in Question 2, and u is the eigenvector of S corresponding to the second largest eigenvalue.
What does the matrix X contain? Derive the expression for the sample covariance of X, in terms of S.
(5 marks)
4. What does it mean for w and u to be orthogonal? Prove that the sample covariance matrix of X is
diagonal when w and u are orthogonal.
(5 marks)
[Hint: Consider expressing the covariance matrix of X as a function of the eigenvalues, λw and λu,
associated with the eigenvectors w and u, respectively.]
1
Questions for ETF5500 students only
5. Let us construct a new variable A, whose (n× 1) vector of observations is constructed by a = 1√
λw
x1
and a new variable B, whose (n× 1) vector of observations is constructed by b = 1√
λu
x2. The terms
xi is the ith column of the matrix X, defined in Question 3; and the terms λw and λu are defined in
Question 4. Let us collect these two new variables into a new (n× 2) data matrix Z = [ a b ]. Prove
that the sample covariance matrix of Z is an identiy matrix. You MUST use the result from Question 4
in your derivation.
(5 marks)
[Hint: Write Z in terms of X, and derive its covariance matrix.]