Using R package software, then copy the answers directly from the R package please. Question 1 The file “air.xls” contains 42 measurements on air-pollution variables recorded at 12:00 noon in the Los...

Using R package software, then copy the answers directly from the R package please.


Question 1

The file “air.xls” contains 42 measurements on air-pollution variables recorded at 12:00 noon in the Los Angeles area on different days.

1. Obtain the sample correlation matrix R.


2. 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.

1. Estimate the factor loading values ?_jk
   and specific variances ?_j
   using the principal component approach.


2. Estimate the above quantities again using maximum likelihood. What is the difference between the solutions of ML and principal component?


3. Perform a varimax rotation of both solutions in (c) and (d). Interpret the results.


4. Calculate the factor scores from the ML estimates by

1. weighted least squares, and


2. regression approach.

Question 2

The US Environmental Protection Agency collected magnesium uptake data, which is included in the file “magnes.xls” . 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.

1. A model is suggested in which magnesium uptake is regressed against the time in a quadratic model:

E(y) = ß₀
+ ß₁
x + ß₂
x²
+ ß₃
z
where z is an indicator variable representing the treatments. Fit this regression model.

1. 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) = ß₀
+ ß₁
x + ß₂
x²
(treatment 1)
= g₀
+ g₁
x + g₂
x²
(treatment 2)
Satisfy the hypothesis H₀: ß₁₌g₁
and ß₂₌g₂


One way of testing this hypothesis is by the following steps:

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


2. Identify a C matrix such that the above hypothesis can be expressed as H₀: Cß = 0.

Clearly specify the matrices X, ß and C

1. Hence test the above hypothesis.

Question 3

The weekly rates of return for five stocks listed on the New York Stock Exchange are given in file “stock.xls”;

1. Construct the sample covariance (or correlation) matrix S and find the sample principal components.

1. Determine the proportion and the cumulative proportion of the total variance explained by each principal component.

1. Construct a SCREE plot of the eigenvalues. Can we summarize the 5 stock variables in less than 5 dimensions?