Filters may be applied to a time series for a variety of reasons. Suppose that a time series consists of a long-term movement, a trend, upon which is superimposed an irregular component. A moving average filter will smooth the series revealing the trend more clearly.
Assume that is the filtered version of the series. Then
The weights of a moving average filters add up to one i.e. . The simplest such filter is the uniform moving average for which:
The gain of such filter is
The Moving Average filter removes a cycle of period n together with its harmonics.