Optimal band-pass filtering and the reliablity of current analysis

This paper shows how existing band-pass filtering techniques and their extensions may be applied to the common problem of estimating current trends or cycles. These techniques give estimates which are “optimal” given the available data, so their standard errors represent a lower bound on what can be achieved with other univariate techniques. Applications to the problems of estimating current trend productivity growth, core inflation and output gaps are considered. These illustrate the different factors which determine how accurately the underlying trend is measured and the degree to which estimated cycles tend to lead or lag the true cycle. Optimal band-pass filtering and the reliablity of current analysis Simon van Norden January 29, 2004 2 1.0 Introduction A common problem in macroeconomics is that of measuring the business cycle, or more generally, of separating long-run trends from short-term movements. A technique which does so may be thought of as a filter, one which is applied to raw economic data prior to analysis. The best known is the Hodrick-Prescott (HP) filter, which has become the benchmark against which all other filters in macroeconomics are compared.1 The HP filter is unabashedly arbitrary; it was proposed and adopted largely on the basis that it gave results which “looked reasonable. ” Its use has since been rationalized as an approximate band-pass filter. A band-pass filter is one which isolates movements in a series between a specified upper and lower frequency or duration; movements outside this desired frequency band are eliminated. As commonly used with quarterly economic data, the HP filter eliminates or greatly reduces most long-run movements in the series while preserving those at roughly business cycle frequencies.2 The result is a detrended series which looks like a business cycle and has served as an agnostic basis for much economic analysis.3 Given its preeminent role and its arbitrary nature, the HP filter has been the focus of much emulation and innovation in recent years. Baxter and King (1999) argue in favour of replacing the HP filter with a more exact band-pass filter, arguing that better results come from using a better approximation. Gomez (2001) and Pollock (2000) propose the use of other ad hoc filters which are commonly used in engineering and which also approximate band-pass filters.4 Pedersen (1998) and Kaiser and Maravall (2001) propose extensions or modifications to the HP filter to improve its performance. However, most of this literature has ignored the application to current analysis. The distinguishing feature of current analysis is that it interprets the most recently available information; the cycle (or trend) of interest is that at the end of the data sample. The dominant focus of the above literature is historical analysis, in which we care mostly about the cycle (or trend) somewhere in the middle of the sample. This distinction is sometimes critical. Most of the analysis in the existing literature is restricted to symmetric filters; to isolate the cycle or trend at time t, such filters use an equal number of observations from before and after t.5 This precludes their use at the end of sample. Other filters are justified 1. Its current popularity stems from its use in the seminal working paper Hodrick and Prescott (1977), finally published as Hodrick and Prescott (1997), although the technique dates from the 1920s. 2. In practice, business cycles are defined to have durations between 6 and 32 quarters. This definition gained popularity after Baxter and King (1999) cited Burns and Mitchell (1947) as characterizing business cycles in this way. It is useful to remember that these numbers are not written in stone; Stock and Watson (1998) quote Burns and Mitchell (1947, p. 3) as stating that business cycles vary in duration “. from more than one year to ten or twelve years. ” 3. The claim that HP-filtered output “looks like” a business cycle is a popular misstatement. Business cycle measurement and analysis, since its infancy, has used the HP filter (or even simpler moving-average filters which produce similar results.) It is therefore probably more accurate to say that the output of these filters have defined what we think of as business cycles. See Morley, Nelson and Zivot (1999). 4. Both examine Butterworth filters, of which they note the HP filter is simply a special case. Optimal band-pass filtering and the reliablity of current analysis Simon van Norden January 29, 2004 3 on the basis of their mid-sample properties, but may behave quite differently at the end of sample.6 This paper considers the filtering problem from the perspective of current analysis. Rather than use ad hoc approximations to band-pass filters, it shows how to construct one-sided band-pass filters which are optimal in a minimum mean-squared-error sense. Unlike the filters mentioned in the above literature, the optimal filter will vary with the properties of the data series to be filtered. While such filters are little known in macroeconomics, they are not new; the annex of this paper reviews the contributions of Koopmans (1974) and Christiano and Fitzgerald (1999). The body of this paper gives an overview of these optimal filters and applies them to three problems of widespread interest; estimating the current output gap, the current trend growth rate of productivity and the current trend rate of inflation. Related to the filtering literature discussed above is another one which examines the reliability of filtered estimates of trends and cycles.7 Since the current analysis filter discussed in this paper minimizes a MSE criterion, we can relate its reliability to those of other filters examined in this literature. It also establishes an upper bound on the accuracy that any such filters can hope to achieve. Because this bound depends in a complicated way on the properties of the data analysed, we investigate the properties of the optimal filter for the three common problems in current analysis. The next section of the paper provides a non-technical overview of band-pass filtering and the optimal one-sided band-pass filter. A derivation and a more detailed discussion of this filter may be found in the technical appendix. Section 3.0 applies the filter to three common macroeconomic problems of measuring trend and cycles and discusses the results. Section 4.0 summarizes the conclusions and suggests avenues for further research. 5. Baxter and King (1999), for example, suggest reserving about five years of data from each end of the sample to provide the necessary leads and lags. 6. The HP filter is a case in point. Comparisons to band-pass filters are based on its symmetric MA representation, which is a limit the filter approaches in the middle of a large sample. Its representation at the end of sample is quite different; see St. Amant and van Norden (1997). 7. Examples of this include Setterfield et al. (1992), Staiger, Stock and Watson (1997), Orphanides and van Norden (1999, 2002) and Cayen and van Norden (2002). Optimal band-pass filtering and the reliablity of current analysis Simon van Norden January 29, 2004 4 2.0 An Optimal Band-Pass Filter for Current Analysis 2.1 Filters: A Primer8 A filter may be thought of as an algorithm for processing a time series in order to get a more meaningful statistic; e.g. the process of averaging measured rainfall at a given location to get “average” rainfall. We can describe this mathematically as