  Several phenomena are available representing market activity: volumes, number
of trades, durations between trades or quotes, volatility - however measured -
all share the feature to be represented as positive valued time series. When
modeled, persistence in their behavior and reaction to new information
suggested to adopt an autoregressive-type framework. The Multiplicative Error
Model (MEM) is borne of an extension of the popular GARCH approach for modeling
and forecasting conditional volatility of asset returns. It is obtained by
multiplicatively combining the conditional expectation of a process
(deterministically dependent upon an information set at a previous time period)
with a random disturbance representing unpredictable news: MEMs have proved to
parsimoniously achieve their task of producing good performing forecasts. In
this paper we discuss various aspects of model specification and inference both
for the univariate and the multivariate case. The applications are illustrative
examples of how the presence of a slow moving low-frequency component can
improve the properties of the estimated models.
