PCA on 500 daily equity return series gives eigenvalues where the first explains 38% and the next four explain 4.1%, 3.0%, 2.2% and 1.9%, and the rest tail off. Give two defensible rules for how many…

Question

PCA on 500 daily equity return series gives eigenvalues where the first explains 38% and the next four explain 4.1%, 3.0%, 2.2% and 1.9%, and the rest tail off. Give two defensible rules for how many components to keep and say what each assumes.

Accepted Answer

Keep the components whose eigenvalues exceed the largest eigenvalue a pure noise matrix of the same shape would produce, or keep the number that minimises reconstruction error on held-out data, and both are defensible while a fixed cumulative variance target is not. The first rule uses the fact that with p series and n observations, a sample covariance built from independent noise has eigenvalues spread over a known band whose upper edge grows with the ratio p/n, so with 500 series and a few years of daily data the noise edge is well above zero and everything below it is sampling artefact. On the stated numbers the first component is far outside any noise band and is the market factor, while components beyond the fifth need checking against that edge before being called factors. The second rule holds out rows, reconstructs them from k components fitted on the rest, and picks the k minimising out-of-sample squared error, which assumes only that the structure is stable across the split. A cumulative variance threshold such as 90% assumes the noise carries no variance, which is false, and on this spectrum it would retain hundreds of components. The scree elbow is a visual version of the same idea and is not reproducible between analysts.