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Multivariate Analysis HW canonical correlation

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1 Data Description

There are monthly returns of International Business Machines, Hewlett-Packard,

Intel Corporation, Merrill Lynch, and Morgan Stanley Dean Witter from 1990M1to 1999M12. 120 observations are log and percentage forms. We think that first

there companies have a similar industrial attribute and also the other do. Thus,

the samples could be divided up into two meaningful parts, technology industry

and financial industry, respectively.

First, we list the definition of canonical correlation and the test statistic as

following:

2 Definition of Canonical Correlation

In practice, we usually use to stand for the correlation between two variables.

But in multivariate case, we can make

U1 = aT1

p11

V1 = bT1

p21

to new variables for analysis. Therefore,

V ar( aT1p11

) = aTxa, V ar( bT1p21

) = bTyb, Cov( aT1p11

, bT1p21

) = aTxyb

are the definitions of symbols.

Then, we maximum the covariance of U1 and V1 subject to the variance of U1 and

V1 equals 1.

Max f(a, b) = Cov(aTx, bTy), s.t. V ar(aTx) = V ar(bTx) = 1

By using Lagrange method, we can get

1x xy1y yxa =

a

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1y yx1x xyb =

b

for subsequent calculation. The canonical correlations are determined by the

eigenvalue of the matrices,

1x xy1y yx

1y yx1x xy

3 Test Statistic

In case that p1 = p2,

2xy =Cov2(x, y)

V ar(x)V ar(y)

If we want to test how many canonical correlations will not be 0 significantly, wecould use the test statistic like as following,

(n 1 0.5(p + q + 1)) lnpi=1

(1 2i )

We could compare it with a 2 distribution with pq degree of freedom. Then, we

give a more general expression just like below,

(n 1 0.5(p + q + 1)) lnp

i=k+1

(1 2i ) > 2 [(p k)(q k)]

4 Data Analysis

We have five companies return rates and want to measure the correlation

between two groups we setted earlier. After calculation, we get these two matrices,

1x xy1y yx

1y yx1x xy

and the canonical correlations are determined by the eigenvalue of these matrices.

4.1 Correlation Matrix

Before we analyse the data, we could observe the Pearson correlation between

variables.

IBM HW P IN T C M ER M W D

IBM 1.0000000

HW P 0.4191051 1.0000000

I N T C 0.2967149 0.4496329 1.0000000

M ER 0.1827630 0.3595049 0.2549587 1.0000000

M W D 0.1821550 0.3815569 0.2399726 0.7934863 1.0000000

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There is a strong correlation between MER and MWD, which is just like we

assumed. A similar relationship can also be observed in technology industry.

4.2 Eigenvalues

By calculating these two matrices, five eigenvalue are shown as following:

Table 1: Eigenvalues

1.631165e-01

2.670369e-03

-1.427730e-19

0.163116536

0.002670369

Then, we can take the square root of the biggest two, 0.163116536 and 0.002670369.

It means that the first canonical correlation, 1, is 0.4038769, and the second, 2,

is 0.05167561.

4.3 2 Statistic Test

We construct two hypotheses,

H0 : 1 = 2 = 0, H1 : 1 or 2 = 0 (1)

H0 : 1 = 0, 2 = 0, H1 : H0 is false. (2)

Table 2: P-valuesp-value 1 p-value 2

0.001860331* 0.8563393

* significant at =0.05 level.

According to the result above, we conclude that the result offers evidence that

support first hypothesis, proving that CC1 is the survived CC in our analysis.

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