Asymmetric filters for trend-cycle estimation

In this chapter, we study the asymmetric filters of GK and L2 from the viewpoints of revision and false turning points. Hence, we derive the asymmetric filters of each of them and analyze their time path, i.e. their behavior in time, by starting with the one applied to the last data point up to the central, symmetric, filter. Since the asymmetric filters are time varying and applied in a moving manner, the estimates based on the most recent observations are subject to revisions as new observations are added. Thus, the real time estimate based on the current and past available observations will be revised six times before the symmetric time invariant filter can be applied. It should be noted that the revisions are due to both filter changes and the innovations introduced by new observations. We shall discuss only the revision due to filter changes. From this viewpoint, we define as `optimal' the smoother with asymmetric filters that satisfy the property of producing revisions that are small in size and monotonically converge to zero. Hence, the time path of the non symmetric filters is here studied in terms of: (1) consecutive filter distances and (2) convergence pattern to the central one. Another important aspect investigated in the chapter concerns the short cycles of 10 months present in the estimated trend, which can lead to the wrong identification of turning points. On this regard, we analyze the power of each asymmetric filter at the frequency $lambda =0.10$ corresponding to cycles of period equal to 10 months, $2pilambdain(0,pi)$. Furthermore, we investigate if one of the non symmetric filters can be used as a substitute for the central one to avoid unnecessary revisions.