A0462
Title: Correlation matrix of factor model: Fluctuation of largest eigenvalue, scaling of bulk eigenvalues, and stock market
Authors: Yohji Akama - Tohoku University (Japan) [presenting]
Abstract: Consider an $N$-dimensional sample of size $T$ and a sample correlation matrix $C$. Suppose that $N$ and $T$ tend to infinity with $T/N$ converging to a fixed finite constant $Q > 0$. If the population is a factor model, then the eigenvalue distribution of $C$ almost surely converges weakly to Marcenko-Pastur distribution such that the index is $Q$ and the scale parameter is the limiting ratio of the specific variance to the $i$-th variable in the limit o4f $i$. For an $N$-dimensional normal population with an equi-correlation coefficient $r$, which is a one-factor model, for the largest eigenvalue $l$ of $C$, we prove that $l/N$ converges to $r$ almost surely. These results suggest an important role of an equi-correlated normal population and a factor model: the histogram of the eigenvalue of the sample correlation matrix of the returns of stock prices fits the density of Marcenko-Pastur distribution of index $T/N$ and scale parameter $1-l/N$. Moreover, the limiting distribution of the largest eigenvalue of a sample covariance matrix of an equi-correlated normal population is provided. The phase transition is discussed regarding the decay rate of the equi-correlation coefficient in $N$.
