2. Theory
2.3 The Taylor Rule
2.3.4 The Hodrick-Prescott Filter
The Hodrick-Prescott filter is a mathematical equation used to distinguish the cyclical component of a data series from the estimated smoothed trend. The filter estimates the trend by calculating a weighted moving average, where the moving average is symmetric and centred. If we assume π! is real GDP in period t, we can define π! as the product of a growth component, π!!, which is the trend value π! would assume if the economy was on its long-term growth path, and a cyclical component, π!!, which fluctuates around a long-run mean value of 1 (SΓΈrensen & Whitta-Jacobsen (2010)).
2.4 π! =π!!π!!
The assumption on the mean value of π!! implies that π! = π!! on average. As we wish to look at percentage change in the variables, it is useful to work with the natural logarithms of
8 BjΓΈrnland, Hilde C., Brubakk, Leif and Jore, Anne Sofie (2008). This paper looks at several ways for estimating potential output and compares the difference in output gap that they produce.
said variables, as change in the log of a variable X approximates percentage change in X.
Log-transforming [2.4] gives us:
2.5 π¦!= π!+π!
Where π¦!= πππ!,π!= πππ!!πππ π! =πππ!! for t=1,β¦,T
In order to determine the growth component, we must separate π! from π!. This is done by solving the following equation with respect to all the π!:
2.6 Min
!!!!!!! π¦!βπ! !
!
!!!
+Ξ» π!!!βπ! β(π!β π!!!) !
!
!!!
The first term of this equation: (π¦!βπ!)!, measures the cyclical component, π!. As neither positive nor negative deviations are desirable, the expression is squared so both types of deviations are weighted the same.
The second term is the moving average multiplied with Ξ», which penalizes the variability in the growth component, π!. As π¦! is measured in logarithms, the terms π!!!βπ! and π!βπ!!! are approximately the percentage growth rates of the trend value of real GDP in periods t+1 and t respectively.
Equation [2.5] provides us with a trade-off between the two components. On the one side we want choose the π! so that the changes in estimated trend is minimized over time. On the other hand, we want to bring π! as close as possible to the log of real output to minimize the first term. The value of the Ξ» will determine the penalizing effect of the second term, and thus the relative weight put on the conflicting objects. Setting π=0 means that the last term becomes insignificant, this would be equal to implying that all fluctuations in π¦! is due to changes in the underlying trend growth9. The other extreme is found by assuming π= β, this would imply that the trend growth is perfectly linear10, meaning trend growth is constant. Thus we see that the smoothness of the trend growth is determined by the π-value.
9 In this case the minimizing problem is solved by setting π¦!=π! for all tΒ΄s.
10 Here the minimizing problem is solved by setting π!!!βπ!=π!βπ!!! for all t=2,3,4β¦.,T-1
Clearly this value should be positive, but finite. The international standard when working with quarterly data is to set π =1.600.
2.3.4.1 Weaknesses of the Hodrick-Prescott Filter
The HP-filter has received its fair share of criticism. One major problem is the preciseness of the estimates at the end-points of a time series. Since the filter uses a weighted moving average, the data from the latest periods are included in the average of an earlier period, and since we do not know the future values of the data series, the scope of the smoothing average determines how close to the present it is possible to estimate the trend11. An implication of this problem is that the estimates of the trend at either end of the time series to a greater degree is affected by the actual output in that period rather than the average for several periods. This problem may be even more prominent when using time data, as some real-time series are the subjects of substantial revisions. This is unfortunate as one is often particularly interested in estimating the output gap for the most recent periods in order to evaluate the need for active macroeconomic policy to smooth the business cycle12.
Another problem is that there is no established βcorrectβ way of determining π. Although there is a common practice of setting π=1600 for quarterly data, this is not unconditionally the best value. This arbitrariness makes using the HP-filter more problematic, as the estimated trend is largely affected by the chosen π-value.
One can also experience problem during particularly long business cycles. If, for instance, the economy is experiencing a prolonged period with a negative output gap, the HP-filter will gradually estimate a lower level of trend growth to close this gap. This may produce a misleading image of the negative output gap closing, which may not truly be the case.
The HP-filter will also not be able to capture structural breaks in the trends of an economic time series. For instance, if the economy experiences a major technological shock, which drastically raises potential production, this will only slowly and gradually be picked up by the HP-filter as the trend level of potential output rises.
11Observations from periods t-1, t and t+1 are used to estimate the trend for period t.
12 Actual monetary policy is also conducted with a forward-looking perspective, thus in order to estimate the optimal policy in real-time, the central bank will rely on estimates of most likely future values for the variables in question. This provides further problems in identifying the optimal policy, as your decisions largely depends on the quality of your estimates.
In order for the filter to work optimally, two conditions must be fulfilled. The first condition being that the original data series must be known to have an I(2) trend. This is essential, or else the filter will create shifts in the trend growth that do not correspond with the original data series. The second condition is that the noise in the original data series is normally distributed. King & Rebelo (1993) argue that none of these conditions are likely to hold true in practice.