The smoothness criterion as a trend diagnostic

The standard approach to time series analysis is to decompose the series into three components, the trend, the seasonal component and the irregular noise. In this paper we are concerned with estimating the trend and so we will assume that all series have been seasonally adjusted. The structural form of the trend is often decided by physical considerations or by inspection of global time series plots. A global parametric model may be used to estimate the trend but a more common approach is to use a finite moving average filter to obtain non-parametric estimates of the local trend. Detailed accounts of recent developments in trend estimation can be found in Kenny and Durbin [5] and Cleveland et al. [1]. Finite symmetric moving average filters for identifying the underlying trend in nonseasonal time series are not new, dating from the seminal work of Henderson [3], in the actuarial literature, and Macaulay [7]. Determining the order of the local polynomial trend is often problematic, particularly if there is subtle local curvature in the presence of a dominant linear trend. The most common diagnostic tool is a sequence of time series plots for the original series and the residual series calculated after each attempt at estimation. In this paper a simple, alternative graphical diagnostic is provided to help select or confirm the order of any local polynomial term in the trend. These plots are motivated by the smoothness criterion used in designing optimal filter weights and appear to be sensitive to local curvature.