Reduced Rank Spatio-Temporal Models

Abstract

To simultaneously model the cross-sectional dependency and dynamic time dependency among n units, most research in spatial econometrics parameterizes the coefficient matrices among the n 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.

KEYWORDS:

• Quasi-maximum likelihood estimator
• Reduced-rank regression
• Ridge-type ratio estimator
• Spatial dynamic panel models

Supplementary Materials

Supplementary material is organized as follows. Section S.1 provides the explicit forms of the first order and second order derivatives of the likelihood function. Section S.2 presents some useful lemmas and their proofs. Section S.3 introduces the proofs of Theorem 1, Corollary 1 and Theorem 2, respectively. Section S.4 presents some discussions for Condition (C5). Section S.5 introduces reduced rank spatio-temporal models with covariates. Section S.6 and Section S.7 introduce the optimization algorithm and some additional simulation studies, respectively. Section S.8 presents the details of the verification for real data.

Acknowledgments

The authors are grateful to the editor, associate editor, and anonymous referees for their insightful comments and constructive suggestions. Kuangnan Fang’s research was supported by the National Natural Science Foundation of China (72071169, 72233002), and the National Social Science Foundation of China (21&ZD146). Wei Lan’s research was supported by the National Key R&D Program of China (2022YFA1003702), National Natural Science Foundation of China (71991472, 12171395, 11931014, 72333001), and the Joint Lab of Data Science and Business Intelligence at Southwestern University of Finance and Economics. Jihai Yu’s research was supported by the National Natural Science Foundation of China (71925006), the Center for Statistical Science of Peking University, China and Key Laboratory of Mathematical Economics and Quantitative Finance (Peking University) of the Ministry of Education, China. Wei Lan and Jihai Yu also gratefully acknowledge the financial support from the National Natural Science Foundation of China (72333001). Qingzhao Zhang’s research was supported by the National Bureau of Statistics of China (2022LZ34), and the National Science Foundation of China (71988101).

Disclosure Statement

No potential conflict of interest was reported by the author(s).