  The parametric estimators applied by rolling are commonly used in the
analysis of time series with nonlinear features, such as structural change due
to time varying parameters and local trends. This paper examines the properties
of rolling estimators in the class of Temporally Local Maximum Likelihood
(TLML) estimators. We study the TLML estimators of constant parameters,
stochastic and stationary parameters and parameters with the Ultra Long Run
(ULR) dynamics bridging the gap between the constant and stochastic parameters.
Moreover, we explore the properties of TLML estimators in an application to the
Susceptible-Infected-Susceptible (SIS) epidemiological model and illustrate
their finite sample performance in a simulation study.
