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An observed $K$-dimensional series $\left\{ y_{n}\right\} _{n=1}^{N}$ isexpressed 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 theobservation. The goal is to determine the dimension of the factors, $p$, thefactor loading matrix, $Q$, and the factors $f_{n}$. Here, it is assumed thatthe noise co-variance is positive definite and allowed to be correlated withthe 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 todetermining the numerical rank of $\tilde{M}$. Using Rank Revealing QR (RRQR)decomposition, a model order detection scheme is proposed for determining thenumerical rank and for estimating the loading matrix $Q$. The rate ofconvergence of the estimates, as $K$ and $N$ tends to infinity, is derived andcompared with that of the existing Eigen Value Decomposition based approach.Two applications of this algorithm, i) The problem of extracting signals fromtheir noisy mixtures and ii) modelling of the S&P index are presented.

A QR Decomposition Approach to Factor Modelling: A Thesis Report