Double Instrumental Variable Estimation of Interaction Models with Big Data

The static factor analysis of a (n, m) matrix of observations Y is based on  the joint spectral decomposition of the matrix squares YY’ and Y’Y for Principal Component Analysis (PCA).  For very large matrix dimensions n and m, this approach has a high level of numerical complexity.  Indeed, the number of required computations grows cubically w.r.t. the matrix dimensions, that is, much faster than the number of observations.  The big data feature can be used to propose new estimation methods with a smaller degree of numerical complexity.  The double Instrumental Variable (IV) approach uses row instruments and column instruments to estimate consistently the factors up to K2 parameters, where K is the number of factors.  Then, the factor model can be concentrated and the K2 missing parameters estimated consistently by Ordinary Least Squares (OLS).  We compare the double IV approach to PCA in terms of numerical complexity and statistical efficiency.  The double IV approach can be used for the analysis of recommender systems and provides a new collaborative filtering approach.