High-Dimensional Binary Variates: Maximum Likelihood Estimation with Nonstationary Covariates and Factors

TL;DR

This paper develops a high-dimensional binary variate model with nonstationary covariates and factors, providing asymptotic theory, convergence rates, and applications to financial market jump analysis.

Contribution

It introduces a novel framework for high-dimensional binary data with nonstationary elements, deriving new asymptotic properties and convergence rates for MLE estimators.

Findings

MLE of coefficients has dual convergence rates under certain conditions

MLE of nonstationary factors is consistent with specific growth conditions

Factors' limiting distributions depend on time, with improved rates in cointegrated cases

Abstract

This paper introduces a high-dimensional binary variate model that accommodates nonstationary covariates and factors, and studies their asymptotic theory. This framework encompasses scenarios where single indices are nonstationary or cointegrated. For nonstationary single indices, the maximum likelihood estimator (MLE) of the coefficients has dual convergence rates and is collectively consistent under the condition $T^{1/2}/N\to0$, as both the cross-sectional dimension $N$ and the time horizon $T$ approach infinity. The MLE of all nonstationary factors is consistent when $T^{\delta}/N\to0$, where $\delta$ depends on the link function. The limiting distributions of the factors depend on time $t$, governed by the convergence of the Hessian matrix to zero. In the case of cointegrated single indices, the MLEs of both factors and coefficients converge at a higher rate of $\min(\sqrt{N},\sqrt{T})$. A distinct feature compared to nonstationary single indices is that the dual rate of convergence of the coefficients increases from $(T^{1/4},T^{3/4})$ to $(T^{1/2},T)$. Moreover, the limiting distributions of the factors do not depend on $t$ in the cointegrated case. Monte Carlo simulations verify the accuracy of the estimates. In an empirical application, we analyze jump arrivals in financial markets using this model, extract jump arrival factors, and demonstrate their efficacy in large-cross-section asset pricing.