Factor Models With Sparse Vector Autoregressive Idiosyncratic Components

We reconcile dense and sparse modelling by exploiting the positive aspects of both. We employ a high-dimensional, approximate static factor model and assume the idiosyncratic term follows a sparse vector autoregressive model (VAR). The estimation is articulated in two steps: (i) factors and loadings are estimated via principal component analysis (PCA); (ii) a sparse VAR is estimated via the lasso on the estimated idiosyncratic components from (i). Step (ii) allows to model cross-sectional and time dependence left after the factors estimation. We prove the consistency of this approach as the time and cross-sectional dimensions diverge. In (ii), sparsity is allowed to be very general: approximate, row-wise, and growing with the sample size. However, the estimation error of (i) needs to be accounted for. Instead of simply plugging-in the standard rates derived for the PCA estimation of the factors in (i), we derive a refined expression of the error, which enables us to derive tighter rates for the lasso in (ii). We discuss applications on forecasting & factor-augmented regression and present an empirical application on macroeconomic forecasting using the Federal Reserve Economic Data - Monthly Database (FRED-MD).