ABSTRACT
A compromise between these two linear extreme approaches is the HP filter (Hodrick and Prescott 1997), which decomposes a nonstationary time series into a nonlinear smooth (slow varying) trend and a cyclic series around the trend, so that the average period of the cyclic series is in the range of NBER (National Bureau of Economic Research) business cycle about two to ten years with an average between four to five years. The HP smooth growth trend [G(t)] is obtained by minimizing the following function:
Min X t G t s G t G t G t G t[ ( ) ( )] {[ ( ) ( )] [ ( ) ( )]}− + + − − − −∑∑ 2 21 1 Here, s is the positive smoothing parameter in the HP filter, which penalizes variability in the growth component series.2 For US annual data, s = 400 for annual data, 1600 for quarterly, and 14,400 for monthly series. The last parameter was suggested by Kydland. LL cycles of XLLc(t) are residuals from log-linear trend. LLg growth trend can be considered as the limiting case of the HPg growth trend when s goes to infinity for logarithmic data. In principle, a choice of observation reference is associated with a theory of economic dynamics. Log-linear detrending implies a constant exponential growth which is the base case in the neo-classical exogenous growth theory (EXGT). The FD detrending produces a noisy picture that is predicted by the geometric random-walk model with a constant drift (or the so-called unit-root model in econometric literature). The efficient market hypothesis simply asserts that stock price movement is a martingale with short correlations in finance theory. Economically speaking, the FD detrending in econometrics implies a mechanical system with only speed without care of its historical position. In other words, the level information in price indicators can be ignored in economic behavior. This assertion contradicts with economic practices, because traders constantly watch economic trends. Most economic contracts, including margin accounts in stock trading, are based on nominal terms. The error-correction model in econometrics tried to remedy the problem by adding some lagged-level information, such as using a one-year-before level as an approximation of the long-run equilibrium (Baba et al. 1992). Few will make an investment decision based only on the current rate of price changes. Then comes the problem of what is the long-run equilibrium in the empirical sense. Option traders based on the Black-Scholes model find that it is extremely difficult to predict the mean, variance, and correlations from historical data (Merton 1990). A proper decomposition of trend and cycles may find an appropriate scheme to weigh short-term and long-run impact of economic movements in economic dynamics. From the view of complex systems, the linear approach is not capable of describing complex patterns of business cycles (Day and Chen 1993). We need a better alternative of detrending. Statistically, a unit-root model can be better described by a nonlinear trend (Bierens 1997). The question is which nonlinear trend is proper for catching the pertinent feature of business-cycle mechanism.
