STAT8121 Assignment 3 Due 6pm on Friday, Oct 30, 2020 Question 1 The US Environmental Protection Agency collected magnesium uptake data, which is included in the file “magnes

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STAT8121 Assignment 3
Due 6pm on Friday, Oct 30, 2020

Question 1
The US Environmental Protection Agency collected magnesium uptake data, which is included in the
file “magnes.xls” available on the unit webpage. Moreover, this data set contains the amount of
magnesium uptake measured at different times with two different treatments. It is anticipated that the
two treatments used may result in different regression equations.
(a) A model is suggested in which magnesium uptake is regressed against the time in a quadratic

model:

E(y) =B, +B, x +B, x +B; Z

where z is an indicator variable representing the treatments. Fit this regression model.
(b) A researcher wants to determine if the simple indicator variable is really appropriate. Basically, the

question is equivalent to whether the two separate models for treatments

E(y)=B, +B, x +B, x? (treatment 1)
=y) +7, X+Y¥>X° (treatment 2)

satisfy the hypothesis Ho: B1=y1 and B2 = y2.

One way of testing this hypothesis is by the following steps

(1) Combine the two models into one E(y) = X B with appropriate design matrix X

and coefficient vector B. In this question B is 6x1.

(2) Identify a C matrix such that the above hypothesis can be expressed as Ho: CB = 0.

Clearly specify the matrices X, B and C.
(c) Hence test the above hypothesis using the T°” test.
(d) Also perform the above test using Full and Reduced models.

Question 2
The file “air.xls” (see the unit web page) contains 42 measurements on air-pollution variables recorded
at 12:00 noon in the Los Angeles area on different days.
(a) Obtain the sample correlation matrix R.
(b) Find eigenvalues and eigenvectors of R. Then determine how many common factors are needed for
the FA model.
We assume the common factor number m = 2 for the following questions.
(c) Estimate the factor loading values ?jk and specific variances ?j using the principal component
approach.
(d) Estimate the above quantities again using maximum likelihood. What is the difference between the
solutions of ML and principal component?
(e) Perform a varimax rotation of both solutions in (c) and (d). Interpret the results.
(f) Calculate the factor scores from the ML estimates by
(1) weighted least squares, and
(2) regression approach.
Question 3
The weekly rates of return for five stocks listed on the New York Stock Exchange are given in file
“stock.xls”; see the unit webpage.
(a) Construct the sample covariance (or correlation) matrix S and find the sample principal components.
(b) Determine the proportion and the cumulative proportion of the total variance explained by each principal component.
(c) Construct a SCREE plot of the eigenvalues. Can we summarize the 5 stock variables in less than 5 dimensions?