Abstract: This paper studies vector autoregressive model (VAR) with interactive fixed effects in high-dimensional setting, which allows both the number of cross-sectional units N and time periods T go to infinity. Assuming that VAR transition matrix is low rank, this paper first proposes nuclear norm regularization based method that estimates transition matrix and interactive fixed effects simultaneously. Under certain conditions, the paper shows that on average, the deviation of each element in the estimated matrix shrinks to 0 as N,T rises. Since nuclear norm penalty induces biases, a debiased procedure is then introduced in order to improve estimators' finite sample performances. Independently, leveraging on principle component analysis, the paper proposes multi-stage estimation method that estimates parameters in multiple stages. I show that the method improves the convergence rates of the VAR transition matrix and reduces biases. In Monte Carlo simulation, I examine estimators' performances in finite sample and the results agree with the theory. Empirically, the paper revisits the US macro data from Mccracken and Ng 2016 and shows the model has great advantage in forecasting macro indexes (IP, CPI and federal funds rate) compared with reduced rank VAR model and pure factor model, especially in long horizon. I also apply the model into analyzing dynamic connectedness among 29 countries using 96 banks' volatility data from Demirer et.al 2018. Results show that during economic crises, latent factors appear and dramatically increase the connectedness level. However, before and after crises, the factors disappear and the connectedness among countries remains at low level.
Estimation of High-Dimensional Seemingly Unrelated Regression Models (Econometric Reviews, 2021)
with Khai X. Chiong and Hyungsik Roger Moon
Abstract: In this paper, we investigate seemingly unrelated regression (SUR) models that allow the number of equations (N) to be large and comparable to the number of observations in each equation (T). It is well known that conventional SUR estimators, for example, the feasible generalized least squares (FGLS) estimator from Zellner [1962] does not perform well in a high dimensional setting. We propose a new feasible GLS estimator called the feasible graphical lasso (FGLasso) estimator. For a feasible implementation of the GLS estimator, we use the graphical lasso estimation of the precision matrix (the inverse of the covariance matrix of the equation system errors) assuming that the underlying unknown precision matrix is sparse. We show that under certain conditions, FGLasso converges uniformly to GLS even when T < N, and it shares the same asymptotic distribution with the efficient GLS estimator when T > N log N . We confirm these results through finite sample Monte-Carlo simulations.