傅里叶递归方法用于潜在马尔可夫模型的估计与推断

中文摘要 AI 辅助

本文针对一类广泛的潜在马尔可夫模型（LMMs）提出了一种新的理论上精确的傅里叶递归框架，该模型涵盖了经济学中广泛使用的各类模型。该框架可视为著名的卡尔曼滤波器在非高斯和非线性LMMs中的对应物。傅里叶系数的闭式递归更新同时实现了滤波、似然评估，并在线累积得分和黑塞矩阵。我们引入了一种统一的截断实现，确保误差控制和数值稳定性，防止近似误差在递归过程中累积。我们建立了LMMs的可行最大似然估计量的渐近性质，提供了一种基于递归的Fisher信息一致估计量，并讨论了可能的模型误设。这些结果为LMMs中的可行近似最大似然估计提供了首个一般渐近理论。模拟实验证明了其准确性和稳定性，而应用于美国破产数据则恢复了一个持续的潜在破产压力过程。

英文摘要

This paper proposes a new theoretically exact Fourier recursion framework for a broad class of latent Markov models (LMMs), encompassing models widely used across a broad range of fields in economics. It can be viewed as a counterpart of the celebrated Kalman filter for non-Gaussian and nonlinear LMMs. Closed-form recursive updates of Fourier coefficients jointly deliver filtering, likelihood evaluation, and simultaneously accumulate the score and Hessian online. We introduce a unified truncated implementation that ensures uniform error control and numerical stability, preventing approximation errors from accumulating through the recursion. We establish asymptotic properties of the feasible maximum likelihood estimator for LMMs, provide a recursion-based consistent estimator for the Fisher information and discuss the possible model misspecification. These results provide the first general asymptotic theory for feasible approximate maximum likelihood estimation in LMMs. Simulations demonstrate its accuracy and stability, while an application to U.S. bankruptcy data recovers a persistent latent bankruptcy-pressure process.

发表机构

• Guanghua School of Management, Peking University（光华管理学院，北京大学）
• Department of Statistics, London School of Economics（伦敦政治经济学院统计系）

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