Mean Square Errors of factors extracted using principal components, linear projections, and Kalman filter

TL;DR

This paper compares the mean square errors of factors extracted using principal components and Kalman filter methods under various assumptions about cross-correlations, highlighting differences in their accuracy and implications for confidence interval construction.

Contribution

It provides a theoretical comparison of the finite N mean square errors of PC and KF factor extraction methods under different correlation structures, revealing their relative performance.

Findings

KF factors have smaller MSEs than PC factors under certain correlation assumptions.

Mis-specification of idiosyncratic components affects the accuracy of factor estimates.

Results inform the construction of confidence intervals for extracted factors.

Abstract

Factor extraction from systems of variables with a large cross-sectional dimension, $N$, is often based on either Principal Components (PC)-based procedures, or Kalman filter (KF)-based procedures. Measuring the uncertainty of the extracted factors is important when, for example, they have a direct interpretation and/or they are used to summarized the information in a large number of potential predictors. In this paper, we compare the finite $N$ mean square errors (MSEs) of PC and KF factors extracted under different structures of the idiosyncratic cross-correlations. We show that the MSEs of PC-based factors, implicitly based on treating the true underlying factors as deterministic, are larger than the corresponding MSEs of KF factors, obtained by treating the true factors as either serially independent or autocorrelated random variables. We also study and compare the MSEs of PC and KF factors estimated when the idiosyncratic components are wrongly considered as if they were cross-sectionally homoscedastic and/or uncorrelated. The relevance of the results for the construction of confidence intervals for the factors are illustrated with simulated data.