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

TL;DR

This paper develops a high-dimensional binary factor model with nonstationary covariates and factors, accommodating both nonstationary and cointegrated single indices. It establishes novel MLE-based asymptotics, including dual convergence rates for coefficient estimators under $I(1)$ dynamics and faster rates under cointegration, with factor estimators exhibiting time-dependent behavior in the nonstationary case and time-invariant behavior under cointegration. A rank-minimization approach selects the number of factors, with consistency guarantees; the limiting distributions differ markedly between the nonstationary and cointegrated regimes. The methodology is validated via Monte Carlo simulations and an empirical application to jump arrivals in financial markets, where extracted jump-arrival factors enhance asset-pricing performance beyond standard FF factors.

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 $T^δ/N\to0$, where $δ$ 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.

Figures (5)

• Figure 1: Estimated number of factors for each year.
• Figure 2: Box plots of factor loadings across different industries. Notes. The three graphs correspond to the three sets of factor loadings, with the horizontal axis representing various industries. The red “+” symbols indicate outliers, and the plots display the confidence interval gaps.
• Figure 3: Estimated factors and their corresponding first-order differences. Note. The top panel displays the three estimated factors, while the bottom panel shows the corresponding first-order difference series.
• Figure 4: Canonical correlations and asset pricing results. Notes. The left panel displays the canonical correlation coefficients between the four jump arrival factors and the Fama-French-Carhart five factors. The middle panel shows the incremental $R^2$ from adding jump arrival factors to the Fama-French-Carhart five-factor model, while the right panel displays the incremental improvement in the joint test statistic (GRS statistic) for alpha.
• Figure 5: Time-variation in the percentage of explained variation for different factors. Note. This figure plots the percentage of explained variation calculated on a moving window of one year (252 trading days).

Theorems & Definitions (16)

• Remark 1
• Definition 1
• Theorem 3.1
• Corollary 3.1
• Theorem 3.2
• Corollary 3.2
• Theorem 3.3
• Corollary 3.3
• Corollary 3.4
• Proposition 1