The classical method of time series decomposition originated in the 1920s and was widely used until the 1950s. It still forms the basis of later time series methods, and so it is important to understand how it works. The first step in a classical decomposition is to use a moving average method to estimate the trend-cycle, so we begin by discussing moving averages.

Notice how the trend (in red) is smoother than the original data and captures the main movement of the time series without all the minor fluctuations. The moving average method does not allow estimates of $T_{t}$ where $t$ is close to the ends of the series; hence the red line does not extend to the edges of the graph on either side. Later we will use more sophisticated methods of trend-cycle estimation which do allow estimates near the endpoints.

The order of the moving average determines the smoothness of the trend-cycle estimate. In general, a larger order means a smoother curve. The following graph shows the effect of changing the order of the moving average for the residential electricity sales data.

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