"Frontmatter". In: Analysis of Financial Time Series

FACTOR ANALYSIS 341(a) 5 stock returns(b) 5 bond index returnseigenvalue0.5 1.0 1.5 2.0 2.5•••••eigenvalue0 1 2 3 4••• • •1 2 3 4 5component1 2 3 4 5componentFigure 8.11. Scree plots for two 5-dimensional asset returns: (a) series of Example 8.7, and(b) bond index returns of Example 8.9.are relatively small and all about the same size, one can determine the appropriatenumber of components. For both plots in Figure 8.11, two components appear to beappropriate. Finally, except for the case in which λ j = 0for j > i, selecting the firsti principal components only provides an approximation to the total variance of thedata. If a small i can provide a good approximation, then the simplification becomesvaluable.8.8 FACTOR ANALYSISOne of the main difficulties in multivariate analysis is the “curse of dimensionality.”In particular, the number of parameters of a parametric model often increasesdramatically when the order of the model or the dimension of the time series isincreased. Simplifying methods are often sought to overcome the difficulty of curseof dimensionality. From an empirical viewpoint, multivariate data often exhibit similarpatterns indicating the existence of common structure hidden in the data. Factoranalysis is one of those simplifying methods available in the literature. The aim offactor analysis is to identify a few factors that can account for most of the variationsin the covariance or correlation matrix of the data.