Reduced Rank Spatio-Temporal Models

To simultaneously model the cross-sectional dependency and dynamic time dependency among <i>n</i> units, most research in spatial econometrics parameterizes the coefficient matrices among the <i>n</i> units as functions of known weights matrices. This modeling framework is over-simplified and faces the risk of misspecification when constructing the weights matrices. In this article, we propose a novel reduced-rank spatio-temporal model by assuming the coefficient matrices follow a reduced-rank structure. This specification avoids construction of the weights matrices and provides a good interpretation, especially for financial data. To estimate the unknown parameters, a quasi-maximum likelihood estimator (QMLE) is proposed and obtained via the Gradient descent algorithm with Armijo line search. We establish the asymptotic properties of QMLE when the number of units and the number of time periods both diverge to infinity. To determine the rank, we propose a ridge-type ratio estimator and demonstrate its rank selection consistency. The proposed methodology is illustrated via extensive simulation studies. Finally, a Chinese stock dataset is analyzed to investigate the cross-sectional and temporal spillover effects among stock returns.