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

Gain of uniform moving average filter

The uniform moving average filter (applied on artificial data)

The Moving Average filter removes a cycle of period n together with its harmonics.

The corresponding **detrended series** is

Hence

, ,

The weights sum up to zero ( thus

and

Bear also in mint that

and

where

is the **seasonal** weight function.

Hence

If is an process with the a.c.g.f. of is given by

with **Spectrum**

where is the spectrum of

and thus we can show that

which for the complement of a simple moving average filter equals

Gain for stationary series with MA and Detrending filter (n=13)

**Example: Detrended Random Walk**

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