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Multivariate Analysis HW 0421PCA

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1 Motivation

Everyone knows that stock markets in a certain geographic region may have

connection with other countries in the same region. Therefore, I choose six coun-tries to verify the concept that I mentioned. These countries all belong to Asia,a rapidly growing area in the world, including Hong Kong(hk), Japan(jp), Ko-rea(kr), Philippines(ph), Singapore(sg) and Taiwan(tw).

2 Data Description

Data Source: Dow Jones Online Database

Period: start: 1993/1/7, end: 2010/12/30

Data Property:

Six Asia countries are as follows: Hong Kong, Japan, Korea, Philip-pines, Singapore, Taiwan

Weekly aggregate index return rate

= ln(close pricet) ln(close pricet1)

Observations: 937

3 Covariance Matrix

First, values in covariance matrix are adjusted to six decimal places for sim-plicity. The biggest value in table 1 is the covariance of the return rate of Korea,which means that the return rate of Korea has a larger fluctuation than othercountries in sample. According to the result we got, it is reasonable that Korea isgoing to be a key role in the following context.

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Table 1: Covariance Matrixhk jp kr ph sg tw

hk 0.001261

jp 0.000397 0.000838kr 0.000886 0.000627 0.002913ph 0.000666 0.000338 0.000610 0.001765sg 0.000799 0.000418 0.000815 0.000753 0.001086tw 0.000716 0.000381 0.000868 0.000585 0.000689 0.001595

Table 2: Correlation Matrixhk jp kr ph sg tw

hk 1.0000000 jp 0.3860739 1.0000000

kr 0.4625107 0.4012562 1.0000000ph 0.4462651 0.2782148 0.2689110 1.0000000sg 0.6828179 0.4379678 0.4581428 0.5437817 1.0000000tw 0.5050966 0.3292490 0.4028976 0.3487346 0.5236347 1.0000000

4 Correlation Matrix

In table 2, only one correlation coefficient exceeds 0.6. It seems that thereis a strong relationship between Hong Kong and Singapore. These two countrieshave some similar characteristics, for instance, a small population size, the type

of city, and a society influenced deeply by Chinese culture. We could assume thatinvestors in these two countries may focus on similar events and respond to marketsimilarly.

5 Eigenvalues and Eigenvectors

Table 3: Eigenvalues and EigenvectorsEigenvalue Proportion

0.005076 0.5366070.001698 0.1795500.001039 0.1098740.000711 0.0751370.000592 0.0625120.000344 0.036312

We calculate the eigenvalue and the proportion each country which ordered intable 1. Obviously, the cumulative proportion of first two countries exceeds 0.7,with a actual value about 0.716. It means that 71.6% of total variance explainedby these PCs. Therefore, we can choose first two weighted equation (principal

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component) from our result and listed as following:

PC1 = 0.381140hk0.227562jp0.619441kr0.365930ph0.363466sg0.391439tw

PC2 = 0.1810871hk+0.008003jp+0.715916kr0.608028ph0.228936sg0.180273tw

The coefficients in PC1 have the same direction, and we could name PC1 gen-eral Asia index that stands for entire Asia stock market. As mentioned in theabove, PC1 gives a higher weight to Korea, meaning Korea is more representativethan other countries.

Then, we see PC2 in the result. Interestingly, the direction of coefficient ofKorea and Japan contrast with other countriess. We might name PC2 regionalfeature regional feature, respectively.

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