Model selection plays an important role in longitudinal data analysis, especially when models are estimated using the generalized method of moments (GMM) in the presence of time-dependent covariates. In this setting, the number of valid moment conditions can grow quickly and may lead to over-parameterized models. The Kullback--Leibler Information Criterion (KLIC) has been proposed as a model-selection tool for this framework; however, the original KLIC criterion may favor overly complex models when the number of parameters or valid moment conditions increases. To address this limitation, this study proposes two penalized versions of KLIC that incorporate penalties based on both the number of model parameters and the number of valid moment conditions. The proposed criteria are referred to as the Moment--Parameter Product Penalty KLIC (MPPP--KLIC) and the Logarithmic Penalty KLIC (LP--KLIC). These criteria provide a theoretically motivated mechanism for balancing model fit and model complexity in GMM-based longitudinal models. Through an extensive simulation study involving both binary and continuous response settings, the proposed criteria are shown to improve the ability of KLIC to distinguish among competing models and to reduce the selection of over-parameterized models. The performance of the proposed methods is further illustrated using the Filipino Child Morbidity dataset, a longitudinal study of child health in the Philippines. The results show that the proposed penalized criteria provide stable and interpretable model rankings and consistently identify age as the most important predictor of child morbidity. Overall, the proposed penalized KLIC criteria offer practical and theoretically grounded tools for model selection in GMM-based longitudinal data analysis with time-dependent covariates.

翻译：模型选择在纵向数据分析中具有重要作用，尤其当模型采用广义矩方法（GMM）估计且存在时变协变量时。在此框架下，有效矩条件的数量可能快速增长，导致过度参数化模型。Kullback-Leibler信息准则（KLIC）已被提出作为该场景下的模型选择工具；然而，当参数或有效矩条件数量增加时，原始KLIC准则可能偏向过于复杂的模型。为解决这一局限，本研究提出两种带罚项的KLIC版本，其罚项基于模型参数数量与有效矩条件数量构建。所提准则分别称为矩-参数乘积罚项KLIC（MPPP-KLIC）和对数罚项KLIC（LP-KLIC）。这些准则为基于GMM的纵向模型在拟合优度与复杂度之间提供了理论驱动的平衡机制。通过涵盖二元与连续响应场景的广泛模拟研究，证明所提准则能提升KLIC区分竞争模型的能力，并减少对过度参数化模型的选择。进一步利用菲律宾儿童发病率数据集（一项关于菲律宾儿童健康的纵向研究）展示了所提方法的性能。结果表明，所提带罚项准则能提供稳定且可解释的模型排序，并一致识别出年龄是儿童发病率的最重要预测因子。总体而言，所提带罚项KLIC准则为含时变协变量的GMM纵向数据分析中的模型选择提供了兼具实用性与理论依据的工具。