Question

Q.5 (26 pts) Consider the monthly log returns, in percentages, of five US stocks from 1990 to 1999 for 120 observations. The stocks are (1) IBM, (2) Hewlett-Packard (HPQ), (3) Intel (INTC), (4) Merill Lynch (MER), and (5) Morgan Stanley Dean Witter (MWD) The eigenvalues and eigenvectors of the sample correlation matrix are given in the R output. > mm=eigen (R) >print(mm$values, digits=3) [1] 2.456 1.145 0.699 0.495 0.205 >print(mm$vectors, digits=3) 1,1] [,2] [,3] [,4] [,5] (1,] 0.342 0.525 0.6906 0.3618 0.0118 [2,] 0.474 0.314 -0.0425 -0.8199 -0.0496 [3,] 0.387 0.405 -0.7166 0.4140 0.0339 [4,) 0.503 -0.481 0.0517 0.1467 -0.7008 [5,] 0.505 -0.481 0.0708 0.0617 0.7107 (a) (10 pts) Answer the following questions. i. Provide interpretations for the first and second principal components. ii. What is the proportion of the total variance of the standardized vari- ables that are accounted for by the first two principal components? iii. After seeing the two principal components what do you think the sample correlation matrix looks like? Which pairs of variables would have high positive correlations, which would have negative correla- tions, which would have moderate or small correlations? iv. How many number of principal components that will adequately de- scribe the variation in this data? And why?

Pls explain carefully!

Answer 1

(iv) From the eigen vector matrix, only the first principal component have eigenvalues greater than 1. Thus, only one PC explains the variation in the data adequately.

(i) The highest eigenvalues in the eigen vector is the principal components. The first principal component is 0.505 and the second principal component is 0.525.

(ii) Proportion of variance of first Principal component is variance of PC1 divided by the total variance. total variance explained for PC1 is 49.12% and for PC2 is 22.9%

(iii) From the eigenvector matrix, PC1 have strong positive correlation with all the five stock companies, PC2 is positively correlated with IBM,HPQ,INTC but negatively correlated with MER and MWD. PC3 is positively correlated with IBM,MER, and MWD but negatively correlated with HPQ AND INTC. PC4 is positively correlated with the stock companies except HPQ. PC5 is positively correlated with IBM, INTC, MWD and negatively correlated with HPQ and MER.