According to theoretical and empirical studies the established opinion is that the trend in the macroeconomic time series is a non-stationary stochastic process. The most frequently approach is to use time series techniques to extract the trend from the data. The most popular and effective approaches include Hodrick-Prescott filter, the band-pass filter or the Beveridge-Nelson decomposition. In this paper the Hodrick Prescott filter will be used.
The trend component can be estimated by many ways. There are two main categories trend-models. The first category include the deterministic models whish are the linear and the polynomial models. The second category consists of the stochastic models(like the Hodrick-Prescott filter).
Some crucial points of these categories are the following:
Its trend-abstraction method is easily applied. This method assumes that the trend component is independent of the rest part of the series and can be estimated by a linear function:
Y t = Î±0 + Î±1*t + ut
where Y t: the time series, t : the trend, ut: the stochastic error term, Î±0 :the constant, Î±1: the slope term
It comes if we add the square of the trend component:
Y t = Î±0 + Î±1*t +Î±2*t+ ut
The trend-polynomial has the following form:
Y t = A + Î£Î±j*tj + ut= A(t)+ ut
It is obvious that the Î±) and Î²) come from the third form.
In these models the the trend component can be estimated by the regression of Y to A(t).The OLS estimations of the systematic part of the A(t) consists the estimation of the trend component while the estimation of the errors consists the cyclical component and other random components.
This method is not reliable, as it requires strict assumptions in the growth rates such as linearity in the growth rates in case Î±) and stability in the acceleration of the growth rate in case Î²). However, these estimation methods assume that the growth rate is important whereas actually it is observed that it is effective to assume that the growth rate is a non-stationary stochastic processes(Hamilton 1989).
2) Stochastic models
Î±)Î¤he first differences
For the abstraction of the trend we calculate the first differences of the time series we are examining. This method requires that the trend component is a time process which contains randomness and is uncorrelated with the cyclical component.(责任编辑：BUG)