Dynamic Factor Models with Sparse VAR Idiosyncratic Components
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We reconcile the two worlds of dense and sparse modeling by exploiting the positive aspects of both. We employ a dynamic factor model and assume the idiosyncratic term follows a sparse vector autoregressive model (VAR) which allows for cross-sectional and time dependence. The estimation is articulated in two steps: first, the factors and their loadings are estimated via principal component analysis and second, the sparse VAR is estimated by regularized regression on the estimated idiosyncratic components. We prove consistency of the proposed estimation approach as the time and cross-sectional dimension diverge. In the second step, the estimation error of the first step needs to be accounted for. Here, we do not follow the naive approach of simply plugging in the standard rates derived for the factor estimation. Instead, we derive a more refined expression of the error. This enables us to derive tighter rates. We discuss the implications to forecasting and semi-parametric estimation of the inverse of the spectral density matrix and we complement our procedure with a joint information criteria for the VAR lag-length and the number of factors. The finite sample performance is illustrated by means of an extensive simulation exercise. Empirically, we assess the performance of the proposed method for macroeconomic forecasting using the FRED-MD dataset.
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