A QR Decomposition Approach to Factor Modelling: A Thesis Report

Abstract

An observed $K$-dimensional series $\left\{ y_{n}\right\} _{n=1}^{N}$ is expressed in terms of a lower $p$-dimensional latent series called factors $f_{n}$ and random noise $\varepsilon_{n}$. The equation, $y_{n}=Qf_{n}+\varepsilon_{n}$ is taken to relate the factors with the observation. The goal is to determine the dimension of the factors, $p$, the factor loading matrix, $Q$, and the factors $f_{n}$. Here, it is assumed that the noise co-variance is positive definite and allowed to be correlated with the factors. An augmented matrix, \[ \tilde{M}\triangleq\left[\begin{array}{cccc} \tilde{\Sigma}_{yy}(1) & \tilde{\Sigma}_{yy}(2) & \ldots & \tilde{\Sigma}_{yy}(m)\end{array}\right] \] is formed using the observed sample autocovariances $\tilde{\Sigma}_{yy}(l)=\frac{1}{N-l}\sum_{n=1}^{N-l}\left(y_{n+l}-\bar{y}\right)\left(y_{n}-\bar{y}\right)^{\top}$, $\bar{y}=\frac{1}{N}\sum_{n=1}^{N}y_{n}$. Estimating $p$ is equated to determining the numerical rank of $\tilde{M}$. Using Rank Revealing QR (RRQR) decomposition, a model order detection scheme is proposed for determining the numerical rank and for estimating the loading matrix $Q$. The rate of convergence of the estimates, as $K$ and $N$ tends to infinity, is derived and compared with that of the existing Eigen Value Decomposition based approach. Two applications of this algorithm, i) The problem of extracting signals from their noisy mixtures and ii) modelling of the S&P index are presented.