Factor Models With Sparse Vector Autoregressive Idiosyncratic Components

Abstract

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).