"Frontmatter". In: Analysis of Financial Time Series

FACTOR ANALYSIS 345matrix P to transform the factor model so that the common factors have nice interpretations.Such a transformation is equivalent to rotating the common factors in them-dimensional space. In fact, there are infinite possible factor rotations available.Kaiser (1958) proposes a varimax criterion to select the rotation that works well inmany applications. Denote the rotated matrix of factor loadings by L ∗ =[l ∗ ij ] andthe ith communality by c 2 i .Define ˜l ∗ ij= l ∗ ij /c i to be the rotated coefficients scaledby the (positive) square root of communalities. The varimax procedure selects theorthogonal matrix P that maximizes the quantityV = 1 k⎡(m∑ ∑⎣ k ( ˜l ij ∗ )4 − 1 k∑ki=1i=1j=1˜l ∗2ij) ⎤ 2⎦ .This complicated expression has a simple interpretation. Maximizing V correspondsto spreading out the squares of the loadings on each factor as much as possible.Consequently, the procedure is to find groups of large and negligible coefficientsin any column of the rotated matrix of factor loadings. In a real application, factorrotation is used to aid the interpretations of common factors. It may be helpful insome applications, but not so in others.8.8.3 ApplicationsGiven the data {r t } of asset returns, the factor analysis enables us to search for commonfactors that explain the variabilities of the returns. Since factor analysis assumesno serial correlations in the data, one should check the validity of this assumptionbefore using factor analysis. The multivariate Portmanteau statistics can be usedfor this purpose. If serial correlations are found, one can build a VARMA modelto remove the dynamic dependence in the data and apply the factor analysis to theresidual series. For many returns series, the correlation matrix of the residuals of alinear model is often very close to the correlation matrix of the original data. In thiscase, the effect of dynamic dependence on factor analysis is negligible.Example 8.8. Consider again the monthly log stock returns of IBM, Hewlett-Parkard, Intel, Merrill Lynch, and Morgan Stanley Dean Witter used in Example 8.7.To check the assumption of no serial correlations, we compute the Portmanteaustatistics and obtain Q 5 (1) = 34.28, Q 5 (4) = 114.30, and Q 5 (8) = 216.78. Comparedwith chi-squared distributions with 25, 100, and 200 degrees of freedom, thep values of these test statistics are 0.102, 0.156, and 0.198, respectively. Therefore,the assumption of no serial correlations cannot be rejected even at the 10% level.Table 8.9 shows the results of factor analysis based on the correlation matrix usingboth the principal component and maximum likelihood methods. We assume that thenumber of common factors is 2, which is reasonable according to the principal componentanalysis of Example 8.7. From the table, the factor analysis reveals severalinteresting findings: