Why You Should Never Use the Hodrick-Prescott Filter. A Comment on Hamilton (The Review of Economics and Statistics, 2018)

Moura, Alban Article Why You Should Never Use the Hodrick-Prescott Filter. A Comment on Hamilton (The Review of Economics and Statistics, 2018) Journal of Comments and Replications in Economics (JCRE) Suggested Citation: Moura, Alban (2024) : Why You Should Never Use the Hodrick-Prescott Filter. A Comment on Hamilton (The Review of Economics and Statistics, 2018), Journal of Comments and Replications in Economics (JCRE), ISSN 2749-988X, ZBW - Leibniz Information Centre for Economics, Kiel, Hamburg, Vol. 3, pp. 1-17, https://doi.org/10.18718/81781.31 This Version is available at: https://hdl.handle.net/10419/295067 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Journal of Comments and Replications in Economics - JCRE Why You Should Never Use the Hodrick-Prescott Filter A Comment on Hamilton (The Review of Economics and Statistics, 2018) Alban Moura* Journal of Comments and Replications in Economics, Volume 3, 2024-1, DOI: 10.18718/81781.31 JEL: B41, C22, E32 Keywords: HP Filter, Hamilton Filter, Business Cycles, Detrending, Filtering, Comment Data Availability: The data and Matlab code to reproduce the results of this comment can be downloaded at JCRE’s data archive (DOI: 10.15456/j1.2024051.1449561565). Please Cite As: Moura, Alban (2024). Why You Should Never Use the Hodrick-Prescott Filter: A Comment on Hamilton (2018). Journal of Comments and Replications in Economics, Vol 3(2024-1). DOI: 10.18718/81781.31 Abstract Hamilton (2018) argues that one should never use the Hodrick-Prescott (HP) filter to detrend economic time series and proposes an alternative approach. This comment reconsiders Hamilton’s case against the HP filter, emphasizing two simple points. First, in the empirical example Hamilton considers, the HP and Hamilton filters yield cyclical estimates with very similar dynamic properties, questioning the notion that one decomposition outperforms the other. Second, there is a mechanical lag in the Hamilton trend, which might cast doubt on the economic plausibility of the trend-cycle decomposition. It follows that the Hamilton filter might not constitute a systematically better alternative to the HP filter. *Banque centrale du Luxembourg, Département Économie et Recherche, 2 boulevard Royal, L-2983 Luxembourg, [email protected]. Acknowledgements: I thank James Hamilton for detailed and helpful feedback on earlier drafts. For useful comments, I also thank the editor (W. Robert Reed) and one anonymous referee, as well as Patrick Fève, Paolo Guarda, Robert Hodrick, Olivier Pierrard, Xiaoxia Shi, and colleagues at the Banque centrale du Luxembourg. The views presented in this paper are personal and should not be reported as those of the Banque centrale du Luxembourg or the Eurosystem. The author declares that he has no material financial interest related to the results reported in this study. Received December 12, 2022; Revised November 17, 2023; Accepted January 09, 2024; Published February 20, 2024. ©Author(s) 2024. Licensed under the Creative Common License - Attribution 4.0 International (CC BY 4.0). 1 A. Moura – Why You Should Never Use the Hodrick-Prescott Filter (Comment). JCRE (2024-1) 1 Introduction In an important paper, Hamilton (2018) argues that one should never use the HP filter, proposed by Hodrick and Prescott (1981, 1997) to decompose a time series into separate trend and cyclical components. Hamilton makes his point in two steps. First, he highlights three drawbacks of the HP filter: (a) It introduces spurious dynamic relations that have no basis in the underlying datagenerating process (DGP). (b) The estimates at the boundaries of the sample are not reliable. (c) Common choices for the smoothing parameter are not supported by the data. Second, he proposes an alternative regression-based strategy, since known as the Hamilton filter, that, he argues, extracts plausible cyclical components while eschewing the pitfalls of the HP filter. According to Hamilton, these elements close the case against using the HP filter. But is the case really closed? Hamilton’s dismissal of the HP filter runs counter to widespread empirical practice. Limitations of the HP filter are well known, but economists still use it for lack of a better alternative. Therefore, the main question is whether the Hamilton filter indeed improves on the HP filter. Using two simple points, this comment argues that the improvement is not so clear. Hamilton (2018) motivates his criticism of the HP filter from a real-world example: the detrending of consumption and stock prices. Both series resemble random walks, but the cyclical components extracted by the HP filter feature complex dynamics. Hamilton considers that these patterns reflect the filter rather than true properties of the data. Surprisingly, Hamilton does not discuss the properties of the cycles extracted by his alternative approach. This comment fills this gap and finds that the Hamilton cycles exhibit persistence and comovements that closely mirror those found in the HP cycles. Thus, the very example Hamilton invokes to dismiss the HP filter actually fails to establish the superiority of his preferred strategy. If the dynamics found in the HP cycles are spurious, then the similar dynamics found in the Hamilton cycles must be equally misleading. On the contrary, if the dynamics found in the Hamilton cycles are authentic, then the similar dynamics found in the HP cycles imply that the HP filter provides at least a reasonable cyclical estimate. This first point is mostly rhetorical and only questions the bite of Hamilton’s empirical criticism of the HP filter. The second point is more substantial: when the original series is persistent, there is a mechanical delay between the data and the estimated Hamilton trend. At a basic level, this is expected: Hamilton defines the trend as an 8-quarter-ahead forecast for quarterly series, so that the trend component reacts to data movements with an automatic two-year gap. While Hamilton does not discuss this timing, this comment shows that it might result in implausible trend-cycle decompositions. For instance, the estimated trend for stock prices rises during most stock-market contractions, before falling abruptly two years after the actual drop in prices, when valuations are already recovering. A similar issue arises when detrending real output and interpreting the trend as potential output: in this case, potential output rises mechanically during most recessions, before sharply falling during the recovery. Of course, given the additive trend-cycle decomposition, questioning the timing of trend estimates necessarily leads to doubts about cyclical estimates as well. In light of these two points, this comment concludes that the Hamilton filter might not offer a systematically superior alternative to the HP filter. A more balanced assessment is that the two filters provide different views of the data, and that which view is more useful is likely to depend on the application. More broadly, the classic question of how to best extract a stationary component from a potentially non-stationary time series remains open and economists can only benefit from 2 Journal of Comments and Replications in Economics - JCRE viewing alternative approaches as complementary rather than substitute, as Canova (1998) argued more than twenty years ago. 1.1 Literature There is an infinite number of ways to detrend a time series, which unsurprisingly led to a large literature comparing, evaluating, and proposing detrending methods. It is beyond the scope of this comment to review this literature. Here, the focus is more narrowly restricted to the HP and Hamilton filters. Hamilton (2018) summarizes the literature about the HP filter. Important papers include Harvey and Jaeger (1993) and Cogley and Nason (1995), who showed that the HP filter can produce cyclical components with dynamic properties absent from the original DGP, and de Jong and Sakarya (2016), who reviewed in detail the econometric properties of the HP filter. Some authors have also evaluated the theoretical properties of Hamilton’s regression filter. For instance, Schuler (2021) studies its spectral properties and shows that it amplifies longer-term cycles and mutes shorter-term fluctuations. As a result, the Hamilton filter typically attributes more medium-term movements to the cycle compared to more standard definitions of the business-cycle phenomenon (see, e.g., Stock and Watson, 1999). Schuler also argues that the Hamilton filter may alter the dynamic relationship between several variables because it induces potentially different phase shifts in the cyclical components. This comment concurs with Schuler that the timing properties of the Hamilton decomposition may be problematic. Jonsson (2020a) documents numerically that the HP and Hamilton filters produce cyclical components with similar dynamic properties in several univariate setups, including the random-walk case. Jonsson concludes that, if one views the properties of HP cycles as problematic, then the same properties found in Hamilton cycles can only be viewed as problematic. This comment complements Jonsson (2020a) at three levels. First, it emphasizes that Jonsson’s critique applies to Hamilton’s own empirical example, questioning his argumentation. Second, it considers a bivariate setup, shedding light on the cross-correlations and joint dynamics discussed by Hamilton (2018). Third, it highlights the timing properties of the Hamilton trend, which affect the economic plausibility of the decomposition. Other authors compare the properties of the two filters when applied to actual or simulated data. For instance, Hodrick (2020) simulates various time-series models approximating the U.S. real gross domestic product (GDP) and evaluates the cyclical components recovered by the HP filter, the Hamilton filter, and a band-pass filter. Hodrick argues that the Hamilton filter outperforms the HP filter for simple DGPs and that the HP and band-pass filters perform better for complex models. Also focusing on real GDP, Hall and Thomson (2021) and Dritsaki and Dritsaki (2022) argue that the HP filter provides more plausible trend-cycle decompositions than the Hamilton filter for New Zealand and Greece. Of course, an issue in interpreting these results is the ambiguity related to the definition of the “true” cyclical component of the data. Another study by Jonsson (2020b) confirms the appealing real-time performance of the Hamilton filter, which behaves better than the HP filter in presence of data revision. 3 A. Moura – Why You Should Never Use the Hodrick-Prescott Filter (Comment). JCRE (2024-1) Finally, a third group of authors propose extensions of the HP and Hamilton filters. For instance, Phillips and Shi (2021) and Mei, Phillips, and Shi (2022) suggest that repeated applications of the HP filter result in improved asymptotic ability to recover a variety of trends. This iterative procedure, dubbed the boosted HP filter, is grounded in the machine-learning theory of boosting. Quast and Wolters (2022), on the other hand, suggest that smoothing the Hamilton trend by averaging across estimates obtained from different forecast horizons leads to cyclical components with better properties. The results reported in this comment indicate that the Quast-Wolters approach does not solve the timing issue associated with Hamilton trends. Lastly, Hamilton and Xi (2023) show how the Hamilton filter can be used to make non-stationary time series amenable to Principal Component Analysis. They also document the robustness of the Hamilton filter to the extreme COVID-19 outliers. 2 The HP and Hamilton Filters For completeness, this section provides a brief characterization of the HP and Hamilton filters and reviews some of their properties. More details can be found in the original papers (Hodrick and Prescott, 1981, 1997; Hamilton, 2018). Both the HP and the Hamilton filters decompose a time series 𝑥𝑡into the sum of two components: 𝑥𝑡=𝑔𝑡+𝑣𝑡, where 𝑔𝑡is the trend and 𝑣𝑡is the cycle. The difference between the two filters lies in the statistical restrictions used to identify the trend component. The HP filter defines the trend component as a smooth variable that does not differ much from the observed series. Given a sample of data {𝑥𝑡}𝑇 𝑡=1, this objective is formalized by choosing 𝑔𝑡as the solution to the following program: min {𝑔𝑡}𝑇 𝑡=−2(𝑇 ∑︁ 𝑡=1 (𝑥𝑡−𝑔𝑡)2+𝜆 𝑇 ∑︁ 𝑡=1 [(𝑔𝑡−𝑔𝑡−1)−(𝑔𝑡−1−𝑔𝑡−2)]2),(1) where 𝜆≥0is a smoothing parameter penalizing large changes in the slope of the trend 𝑔𝑡. The HP trend reduces to the original series when there is no smoothness penalty (𝜆→0) and it corresponds to a linear time trend when the penalty is extreme (𝜆→ ∞). At each period, the HP cycle verifies 𝑣𝑡=𝑥𝑡−𝑔𝑡. Looking at the minimization program (1), it is clear that the value of the HP trend at any given date depends on the full set of available observations on 𝑥𝑡. This can be formalized in two ways. Given a finite sample of data, stacking all observations on 𝑥𝑡in a column vector 𝑥=(𝑥𝑇, 𝑥𝑇−1, . . . , 𝑥1)′allows expressing the HP trend as a column vector 𝑔=𝐴★(𝜆)𝑥, where 𝑔=(𝑔𝑇, 𝑔𝑇−1, . . . , 𝑔−1)′stacks the trend values and 𝐴★(𝜆)is a (𝑇+2, 𝑇)matrix whose entries are functions of the smoothing parameter 𝜆. In population, the HP trend admits a symmetric two-sided representation, 𝑔𝑡=ℎ★(𝐿)𝑥𝑡, where 𝐿is the lag operator and where the filter weights {ℎ★ 𝑗}∞ 𝑗=−∞ are determined only by the value of 𝜆. Given the additive trend-cycle decomposition, there exist similar matrix and filter representations for the HP cycles.1 1One can force the trend and cyclical components at each period to load only on current and past observations of the data. This comment does not discuss this one-sided HP filter. 4 Journal of Comments and Replications in Economics - JCRE Hodrick and Prescott select the value of the smoothing parameter based on prior assumptions about the relative volatility of the trend and cyclical components for typical macroeconomic time series. This leads them to advocate the use of 𝜆=1,600 for quarterly data. Ravn and Uhlig (2002) show how to extend the logic to other frequencies. Turning to the Hamilton filter, it defines the trend component as the value that we would expect for the original series at date 𝑡, based on its behavior up to date 𝑡−ℎ. This is formalized using a simple linear regression of 𝑥𝑡on a constant, the realization ℎperiods ago 𝑥𝑡−ℎ, and 𝑝−1additional lags 𝑥𝑡−ℎ−1,...,𝑥𝑡−ℎ−𝑝+1. For quarterly time series, Hamilton (2018) suggests using ℎ=8quarters and 𝑝=4lags, so that the regression has the following form: 𝑥𝑡=𝑏0+𝑏1𝑥𝑡−8+𝑏2𝑥𝑡−9+𝑏3𝑥𝑡−10 +𝑏4𝑥𝑡−11 +𝑢𝑡.(2) The fitted values and residuals from this linear regression correspond to the estimated Hamilton trend and cycle: 𝑔𝑡=b𝑥𝑡and 𝑣𝑡=b𝑢𝑡. Contrasting the two filters highlights the drawbacks Hamilton and others find in the HP filter. First, the HP filter performs the trend-cycle decomposition using a matrix/filter that depends solely on the smoothing parameter 𝜆, and not on the properties of the series under consideration. For instance, given a value for 𝜆, the HP filter will detrend a white noise and a random walk with the same exact filter. Second, the HP filter is two-sided, so that the trend and cyclical estimates at any given period depend on past, present, and future values of the original series. In finite samples, this implies that the filtered values at the bounds of the sample are defined differently from those in the middle. Third, the value of the smoothing parameter 𝜆is typically chosen without reference to the observed features of the data. Hamilton designs his alternative approach so as to avoid these drawbacks. His regression filter estimates a population property of the DGP, the linear regression of the variable on a constant and its past values, avoiding the use of a fixed detrending operator as done by the HP filter. Simply put, the Hamilton cycle recovers a feature of the original DGP, while the HP cycle is a feature of the filtered DGP. The Hamilton filter’s one-sided nature also ensures that the trend-cycle decomposition at date 𝑡solely relies on the information available up to that date.2Finally, the regression coefficients are estimated from the data, instead of being imposed as in the HP filter. 3 Cyclical Dynamics of Stock Prices and Consumption Section III.A in Hamilton (2018) illustrates the issues arising when one applies the HP filter to detrend typical economic time series using an empirical example, based on stock prices and consumption. This section reexamines this example by submitting the Hamilton filter to the same evaluation as the HP filter. Figures 1 and 2 below reproduce Hamilton’s Figures 2 and 3 using an extended sample. Data definitions and sources are the same as in Hamilton (2018). Stock prices are measured as 100 times the natural log of the end-of-quarter value for the S&P 500 composite stock price index published 2This is a population statement. Given a finite sample of data, all observations contribute to estimating the regression coefficients, so that each period’s trend-cycle decomposition relies on a small amount of future information. 5 A. Moura – Why You Should Never Use the Hodrick-Prescott Filter (Comment). JCRE (2024-1) -0.5 0 0.5 1Stock prices with own lags 0 5 10 Lags -0.5 0 0.5 1Consumption with own lags 0 5 10 Lags -0.5 0 0.5 1 Stock prices with lags of Consumption 0 5 10 Lags -0.5 0 0.5 1 Consumption with lags of Stock prices 0 5 10 Lags -0.5 0 0.5 1Stock prices with own lags 0 5 10 Lags -0.5 0 0.5 1Consumption with own lags 0 5 10 Lags -0.5 0 0.5 1 Stock prices with lags of Consumption 0 5 10 Lags -0.5 0 0.5 1 Consumption with lags of Stock prices 0 5 10 Lags Figure 1: Autocorrelations and cross-correlations for the first differences of stock prices and real consumption. Notes: Upper left: Autocorrelations of the first difference of end-of-quarter value for log S&P 500 composite stock price index. Upper right: Autocorrelations of the first difference of log real consumption. Lower panels: Cross-correlations. by Robert Shiller, available online from http://www.econ.yale.edu/~shiller/data.htm. Consumption is measured as 100 times the natural log of real personal consumption expenditures from the U.S. National Income and Product Accounts. The data are quarterly and run from 1950Q1 to 2019Q4. Figure 1 reports the autocorrelation structure for the first differences of log stock prices and real consumption, as well as their cross-correlations. The top panels show that growth in either series is essentially unpredictable, while the bottom panels indicate that after first differencing neither series has strong predictive power for the other. These features are in line with the idea that both variables resemble random walks. Figure 2 reports the same statistics for the HP cycles extracted from the two series when the smoothing parameter takes the standard value 𝜆=1,600. Hamilton emphasizes the presence of a rich auto-regressive structure in the cyclical components of stock prices and real consumption. The cycles are strongly persistent, so that they are predictable from their past values. The crosscorrelations also indicate that the two cycles forecast each other. This discrepancy between the original properties of the data and those of the HP cycles embodies Hamilton’s claim that the HP filter distorts the series: “The rich dynamics in [the cyclical components] are purely an artifact of the filter itself and tell us nothing about the underlying data-generating process. Filtering takes us 6 Journal of Comments and Replications in Economics - JCRE -0.5 0 0.5 1Stock prices with own lags 0 5 10 Lags -0.5 0 0.5 1Consumption with own lags 0 5 10 Lags -0.5 0 0.5 1 Stock prices with lags of Consumption 0 5 10 Lags -0.5 0 0.5 1 Consumption with lags of Stock prices 0 5 10 Lags -0.5 0 0.5 1Stock prices with own lags 0 5 10 Lags -0.5 0 0.5 1Consumption with own lags 0 5 10 Lags -0.5 0 0.5 1 Stock prices with lags of Consumption 0 5 10 Lags -0.5 0 0.5 1 Consumption with lags of Stock prices 0 5 10 Lags Figure 2: Autocorrelations and cross-correlations for HP-filtered stock prices and real consumption. Notes: Upper left: Autocorrelations of HP-filtered end-of-quarter value for log S&P 500 composite stock price index. Upper right: Autocorrelations of HP-filtered log real consumption. Lower panels: Cross-correlations. Smoothing parameter: 𝜆=1,600. from the very clean understanding of the true properties of these series [...] to the artificial set of relations [found in the cycles, which] summarize the filter, not the data.” According to Hamilton, two characteristics of the HP filter combine to generate these spurious dynamics. First, because the HP filter is two-sided, the estimate at each date loads on past, present, and future shocks. It follows that the cyclical component “is both highly predictable (as a result of the dependence on [lagged shocks]) and will in turn predict the future (as a result of dependence on future [shocks]).” Second, the coefficients relating the cyclical estimate to the underlying shocks “are determined solely by the value of 𝜆,” so that the HP filter effectively imposes dynamics on the data. As noted above, Hamilton overcomes these deficiencies by designing his detrending method as an estimated backward-looking regression. Because the coefficients 𝑏0, . . . , 𝑏4in (2) are estimated from the data, the filter adapts to the underlying DGP. Because the regression uses only past information, the estimated trend and cyclical components will not depend on future shocks. Surprisingly, Hamilton (2018) does not report the autocorrelation function for the cycles extracted from stock prices and real consumption by his alternative approach. Yet, evaluating both filters on the same dataset would be a fair comparison. It would also clarify how moving from the two-sided, calibrated HP filter to the one-sided, estimated Hamilton filter affects the cyclical dynamics extracted from the data. Figure 3 fills this gap. Following Hamilton’s recommendation 7 A. Moura – Why You Should Never Use the Hodrick-Prescott Filter (Comment). JCRE (2024-1) -0.5 0 0.5 1Stock prices with own lags 0 5 10 Lags -0.5 0 0.5 1Consumption with own lags 0 5 10 Lags -0.5 0 0.5 1 Stock prices with lags of Consumption 0 5 10 Lags -0.5 0 0.5 1 Consumption with lags of Stock prices 0 5 10 Lags -0.5 0 0.5 1Stock prices with own lags 0 5 10 Lags -0.5 0 0.5 1Consumption with own lags 0 5 10 Lags -0.5 0 0.5 1 Stock prices with lags of Consumption 0 5 10 Lags -0.5 0 0.5 1 Consumption with lags of Stock prices 0 5 10 Lags Figure 3: Autocorrelations and cross-correlations for Hamilton-filtered stock prices and real consumption. Notes: Upper left: Autocorrelations of Hamilton-filtered end-of-quarter value for log S&P 500 composite stock price index. Upper right: Autocorrelations of Hamilton-filtered log real consumption. Lower panels: Cross-correlations. Regression parameters: 𝑝=4and ℎ=8. for quarterly series, the filter uses 𝑝=4and ℎ=8, so that the cyclical components are obtained by regressing each series at date 𝑡on the four most recent observations available at date 𝑡−8. A striking finding is that the Hamilton cycles display virtually the same dynamic behavior as the HP cycles: the cyclical components are very persistent (the autocorrelations decay slowly toward zero); they have strong forecasting power for each other (the cross-correlations are high at several lags); and there are complex dynamics in cross-correlations that are very similar to those found in HP cycles. Observing the magnitude of the correlations, we see that the Hamilton cycles display even more persistence and more cross-variable predictability than the HP cycles. This is confirmed by the business-cycle statistics reported in Table 1: the first-order autocorrelations of Hamilton-filtered series are 0.89 for stock prices and 0.90 for real consumption, larger than the corresponding values computed from HP-filtered series (0.76 and 0.81). Of course, the HP and Hamilton cycles extracted from stock prices and real consumption are different, but the important point is that they share very similar dynamics.3 These are surprising results, which weaken Hamilton’s case against the HP filter. Given Hamil3For instance, Table 1 shows that the Hamilton cycles are about twice as volatile as the HP cycles, even though they have similar persistence properties. 8 Journal of Comments and Replications in Economics - JCRE References Baxter, M., & King, R. G. (1999). “Measuring Business Cycles: Approximate Band-Pass Filters For Economic Time Series.” The Review of Economics and Statistics, 81(4): 575–593. DOI: 10.1162/003465399558454. Canova, F. 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