Analysis of Financial Time Series, Third Edition

9.6 Asymptotic Principal Component Analysis

So far, our discussion of PCA assumes that the number of assets is smaller than the number of time periods, that is, k < T. To deal with situations of a small T and large k, Conner and Korajczyk (1986, 1988) developed the concept of asymptotic principal component analysis (APCA), which is similar to the traditional PCA but relies on the asymptotic results as the number of assets k increases to infinity. Thus, the APCA is based on eigenvalue–eigenvector analysis of the T × T matrix

where is the T-dimensional vector of ones and with being the sample mean of the ith return series. Conner and Korajczyk (1988) showed that as k → ∞ eigenvalue–eigenvector analysis of is equivalent to the traditional statistical factor analysis. In other words, the APCA estimates of the factors are the first m eigenvectors of . Let be the m × T matrix consisting of ...