Question

Stock-price data (weekly rate of return)

Week	Allied Chemical	Du Pont	Union Carbide	Exxon	Texaco
1	0.000000	0.000000	0.000000	0.039473	0.000000
2	0.027027	-0.044855	-0.003030	-0.014466	0.043478
3	0.122807	0.060773	0.088146	0.086238	0.078124
4	0.057031	0.029948	0.066808	0.013513	0.019512
5	0.063670	-0.003793	-0.039788	-0.018644	-0.024154
6	0.003521	0.050761	0.082873	0.074265	0.049504
7	-0.045614	-0.033007	0.002551	-0.009646	-0.028301
8	0.058823	0.041719	0.081425	-0.014610	0.014563
9	0.000000	-0.019417	0.002353	0.001647	-0.028708
10	0.006944	-0.025990	0.007042	0.041118	-0.024630
...	...	...	...	...	...
91	-0.044068	0.020704	-0.006224	-0.018518	0.004694
92	0.039007	0.03854	0.024988	-0.028301	0.032710
93	-0.039457	-0.029297	-0.065844	-0.015837	-0.045758
94	0.039568	0.024195	-0.006608	0.028423	-0.009661
95	-0.031142	-0.007941	0.011080	0.007537	0.014634
96	0.000000	-0.020080	-0.006579	0.029950	-0.004807
97	0.021429	0.049180	0.006622	-0.002421	0.028985
98	0.045454	0.046375	0.074561	0.014563	0.018779
99	0.050167	0.036380	0.004082	-0.011961	0.009216
100	0.019108	-0.033303	0.008342	0.033898	0.004566

using principal components, is the stock rates-of-return data can be summarized in fewer than five dimensions? Explain

Answer 1

Yes, it is possible to summarize the data by projecting it into fewer dimensions. In this case please use any statistical software like SPSS or R.

The eigen values from the correlation matrix from the data set can be evaluated as:

3.0718097, 0.8339257, 0.5030298, 0.3464342, 0.2448006

Sum of the eigen values is 5.

We can see, first three eigen values accumulate to ((3.0718097 + 0.8339257 + 0.5030298)/5)*100 = 88.18%.

Hence, the whole 5-dimensional data set can be projected to a 3-dimensional data with 88.18% accuracy.

Following R-script (just to run in any R console) can do this principal component analysis.

-----------------------------------------------------------------------------------------------------------

## Set the working directory into correct folder to import the data set.

# In working directory save the data as .csv file with name 'Stock Price Data'.
dat <- read.csv('Stock Price Data.csv')[-1]

## Find the correlation matrix.
S <- round(cor(dat), 3)

## Find the loading matrix to get three components which are individual linear combinations
# of five original variables.
load <- t(as.matrix(eigen(S)$vectors[, c(1,2,3)]))
load

## Calculate the new projected data set in three dimension.
new <- list()
for(i in 1:nrow(dat)){
new[[i]] <- t(load %*% t(as.matrix(dat[i, ])))
}
new.dat <- do.call('rbind', new)

colnames(new.dat) <- c('Y1', 'Y2', 'Y3')
new.dat

---------------------------------------------------------------------------------------------------------

See the newly projected data set as 'new.dat'.

If the variables are taken as:

X1 = Allied Chemical

X2 = Du Pont

X3 = Union Carbide

X4 = Exxon

X5 = Texaco

From the loading matrix 'load', it can be noticed, the three new projected variables Y1, Y2, Y3 can be evaluated as:

Y1 = -0.4297153*X1 -0.4634440*X2 -0.50550704*X3 -0.3209922*X4 -0.49192587*X5

Y2 = 0.2810179*X1 + 0.3851993*X2 -0.06884783*X3 -0.8756546*X4 + 0.03375592*X5

Y3 = 0.8501618*X1 -0.2965287*X2 -0.34006214*X3 + 0.1607006*X4 -0.21869750*X5

Y1, Y2, Y3 construct the new projected data.

Further:

The whole data can be projected into 4-dimension also with accuracy of

((3.0718097 + 0.8339257 + 0.5030298 + 0.3464342)/5)*100 = 95.1%

In this case R-script will be:

----------------------------------------------------------------------------------------------------------

## Find the loading matrix to get four components which are individual linear combinations
# of five original variables.
load <- t(as.matrix(eigen(S)$vectors[, c(1,2,3,4)]))
load

## Calculate the new projected data set in four dimension.
new <- list()
for(i in 1:nrow(dat)){
new[[i]] <- t(load %*% t(as.matrix(dat[i, ])))
}
new.dat <- do.call('rbind', new)
colnames(new.dat) <- c('Y1', 'Y2', 'Y3', 'Y4')
new.dat

---------------------------------------------------------------------------------------------------------

In this case four projected variables will be

Y1 = -0.4297153*X1 -0.4634440*X2 -0.50550704*X3 -0.3209922*X4 -0.49192587*X5

Y2 = 0.2810179*X1 + 0.3851993*X2 -0.06884783*X3 -0.8756546*X4 + 0.03375592*X5

Y3 = 0.8501618*X1 -0.2965287*X2 -0.34006214*X3 + 0.1607006*X4 -0.21869750*X5

Y4 = 0.01077335*X1 -0.6106148*X2 -0.03688498*X3 -0.2331131*X4 + 0.75586462*X5

Y1, Y2, Y3, Y4 construct the new projected data.

N.B. Values may differ slightly during calculation, as this data contains only 20 rows and all calculations have been done on the basis of this 20x6 matrix. Although the results will be the same and exactly same decisions can be made from the original data.