Practical Issues

Two practical issues should be resolved to implement this strategy: choice of a proper decom- position method and selection of a specific level of the parameter of the method. As regards the first issue, the Hodrick-Prescott filter (HP filter) is preferred in that the filter has been

popularly used not only by business cycle researchers to extract cyclical components from the various macroeconomic variables but also by researchers in different fields, mainly the ECB economists, to identify the boom and bust periods of asset or housing markets (Bordo and Jeanne, 2002; Detken and Smets, 2004; Adalid and Detken 2007; Goodhart and Hofmann, 2008; Alessi and Detken, 2009; Agnello and Schuknecht, 2011).42 However, it is still question- able whether application of the HP filter to the time series of housing prices is appropriate since the filtering method was originally devised for the time series containing long-term growth components. Apparently, extending the HP filter coverage beyond the boundary of the busi- ness cycle context seems to have been initiated by Gourinchaset al (2001) in which the trend of credit-to-GDP ratio is estimated by the HP filter. The application of the HP filter in the context of asset markets is based on the same assumption that “aggregate economic variables in capitalist economies experience repeated fluctuations about their long-term growth paths” as stated in Hodrick and Prescott (1981). In light of this, before applying the filter to the time series of house prices, the question should be tackled whether the time series shows a persistent growth path similar to that of consumption and investment. The data on housing prices of the Case & Shiller Index in the U.S. is set out in Figure 2.5. The upward linear time trend in the figure implies that certain forces have propelled the time series upward for the last two decades. Based on the graph appearing in Figure 2.5, applying the filter to house prices is compatible with the presumption of Hodrick and Prescott (1981).43

The second issue in operating the filter involves selecting the level of the smoothing param- eter which determines the smoothness of the growth component series obtained by solving

42_{Moving average is another popular method to extract trend components. Borio and Lowe (2002) implicitly}

use average growth rate plus its certain level of standard deviation as the trend.

43

It is questionable whether continuing upward trend also can be discerned if the time horizon of house prices extend further back into the past. For the U.S. case, Shiller (2005) argues long-run upward trend appears not to exist by presenting the longer series of house price index during the period from 1980 to 2004 constructed by combining 5 source indexes (pp.12-25). However, another query can be raised about effectiveness of the constructed index since the range of samples and compilation methods vary depending on the source index.

Figure 2.5. Linear Time Trend of House Prices in U.S.

Data Source: Standard and Poor’s Case & Shiller Index

the following problem.

min {gt}Tt=−1 nXT t=1 ct2+λ T X t=1 [(gt−gt−1)−(gt−1−gt−2)]2 o (2.22)

under the assumption that

yt=gt+ct t= 1,· · ·, T (2.23)

whereytdenotes a given time series,gtare growth components andctare cyclical components

which measure the deviation of the time series from gt. If the parameter λ in the equation

(2.23) approaches infinity, the solution for the minimization problem is simply a linear time trend. Withλ=0, the solution is simply the given original time series.44 Hodrick and Prescott (1981) selectλ=1,600 based on their prior belief about the moderate level of volatility of the growth and cyclical components. In order for the estimated growth component gt to be a

solution to the minimization problem under a set of assumptions,√λis required to equalσ1/σ2 44_{Dejong and Dave (2007) interpret the parameterλas a weight or value given to the smoothness of the growth}

where the numerator and denominator denote the standard error of cyclical components and the second differences of the growth components respectively. The prior of the two standard errors are 5 forσ1 and 1/8 for σ2. These values lead to the ratio of 40.45

Thereafter the value 1,600 for the smoothing parameter has been accepted as the standard level for quarterly business cycle data, 100 for annual data and 14,400 for monthly data (Favero, 2001; pp.54). In stark contrast with the careful considerations on the choice of λ

in the business cycle literature, any rationale for selecting a certain level of the parameter is not explicitly provided in the empirical studies on assets and housing markets listed above. For annual data, Detken and Smets (2004) choose 1,000 for annual data simply following the lead of Gourinchas et al (2001) which again lacks explanation about the reason for choosing that level. On the other hand, Agnello and Schuknecht (2011) select 10,000 for annual data. In all the other remaining papers in the above list of ECB papers, the parameter value is set as 100,000 for quarterly data which by far exceeds the level widely used in the research on business cycle.

Focusing only on the frequency of the data, it is plausible to set the level of λ as 1,600 since the quarterly data will be used in the regression analysis below as in the business cycle research. However, considering the different property of house price data from macroeco- nomic aggregate variables, three values of the parameter, 1,600, 10,000 and 100,000 are tried preliminarily. Judging from the regression results, the level of 1,600 turns out to be the most appropriate one for eliciting a meaningful relationship between the deviation of house prices from their trend and the candidate determinants.