  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.
