We develop a novel asymptotic theory for local polynomial (quasi-) maximum-likelihood estimators of time-varying parameters in a broad class of nonlinear time series models. Under weak regularity conditions, we show the proposed estimators are consistent and follow normal distributions in large samples. Our conditions impose weaker smoothness and moment conditions on the data-generating process and its likelihood compared to existing theories. Furthermore, the bias terms of the estimators take a simpler form. We demonstrate the usefulness of our general results by applying our theory to local (quasi-)maximum-likelihood estimators of a time-varying VAR's, ARCH and GARCH, and Poisson autogressions. For the first three models, we are able to substantially weaken the conditions found in the existing literature. For the Poisson autogression, existing theories cannot be be applied while our novel approach allows us to analyze it.

翻译：我们针对一大类非线性时间序列模型中时变参数的局部多项式（拟）极大似然估计量，发展了一种新颖的渐近理论。在较弱的正则性条件下，我们证明了所提出的估计量是一致的，并且在大样本下服从正态分布。与现有理论相比，我们的条件对数据生成过程及其似然函数施加了更弱的光滑性和矩条件。此外，估计量的偏项具有更简洁的形式。通过将我们的理论应用于时变VAR、ARCH和GARCH以及泊松自回归模型的局部（拟）极大似然估计量，我们展示了这些一般结果的有效性。对于前三个模型，我们能够显著弱化现有文献中的条件。对于泊松自回归模型，现有理论无法适用，而我们的新方法使其得以分析。