A QR decomposition approach to factor modelling

An observed K-dimensional series {yn}n=1N is expressed in terms of a lower p-dimensional latent series called factors fn and random noise εn. The equation, yn=Qfn+εn 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 fn. Here, it is assumed that the noise co-variance is positive definite and is allowed to be correlated with the factors. This paper proposes the use of QR decomposition instead of the standard Eigenvalue Decomposition (EVD) for determining the model order p and the loading matrix Q. Estimation of the model order p is formulated as a Numerical Rank determination problem. Rank Revealing QR (RRQR) decomposition is used for estimating the loading matrix Q. The asymptotic performances of the estimates of p,Q and fn are analyzed by letting K,N→∞. The asymptotic rates, and empirical results, suggests that the proposed technique is both computationally efficient and accurate.