Set Y t the time series that can be written as:
Y t = Y t-1 + et
Where Y t-1 is the trend component and et is the error term of the series
If the real curve of the trend is not linear then the first differences are not enough for the trend abstraction. In addition, if the series is exponential we can apply the logarithms in order to transform it to linear.
Î²) Hodrick-Prescott filter
This method developed by Hodrick and Prescott (1980) and its scope is to extract the stochastic trend from GDP data.
The Hodrick-Prescott filter(HP filter) is used to demonstrate a nonlinear representation of a time series and it is more volatile to long-run fluctuations than short-run. The adjustment of the trend-sensitivity in short-run fluctuations is achieved with the modification of the multiplier Î».This method has some common points with the MA method but is estimated differently.Particularly, it produces a new time series which is as close as possible to the initial and the new series is as smoothest as it can be. This new time series is an estimation of the 'trend'.