hpfilter: An R Implementation of the One- and Two-Sided Hodrick-Prescott Filter

The Hodrick-Prescott Filter

The Hodrick-Prescott filter is a technique commonly used to smooth macroeconomic data such as the GDP. It consists in separating short-term, cyclical movements in the data from the long-term trend. Although the HP filter has received numerous critiques, recent studies show that alternative methods do not necessarily perform better. Furthermore, the Basel committee, an international body responsible for setting international guidelines for prudential regulation, has recommended the usage of the one-sided HP filter for calculations pertaining to the analysis of financial cycles.

Short mathematical overview

Filtering techniques, such as the HP filter are based on the concept of time series decomposition. This means that an original series \(y\) is separated into its trend and cyclical components, with the remainder of the variability being classified as noise (or the error term \(ε_t\)).

\(y_t = ytrend_t + ycycle_t + \varepsilon_t\)

where:

\(t = [0, T]\)

\(T\) - the total number of observations

\(ytrend_t\) - the trend component

\(ycycle_t\) - the cyclical component

\(\varepsilon_t\) - the error term.

The HP filter finds a trend series that solves the minimization problem shown in the equation below:

\(\min\limits_{ytrend} ( \sum\limits_{t=1}^T(y_t-ytrend_t)^2+\lambda*\sum\limits_{t=2}^{T-1}[(ytrend_{t+1}-ytrend_{t})-(ytrend_{t}-ytrend_{t-1})]^2)\)

From the first-order conditions, we derive the following expression to be solved:

\(A*ytrend=y\)

where:

\[ A = \begin{bmatrix}1+\lambda &-2*\lambda &\lambda &0&\cdots&\cdots&\cdots&\cdots&0\\-2*\lambda&1+5*\lambda&-4*\lambda&\lambda&0&\cdots&\cdots&\cdots&0\\\lambda&-4*\lambda&1+6*\lambda&-4*\lambda&\lambda&0&\cdots&\cdots&0\\0 &\lambda &-4*\lambda & 1+6*\lambda & -4*\lambda &\lambda &0 &\cdots&0 \\\vdots&&&&&&&&\vdots\\0&\cdots&\cdots&0 &\lambda &-4*\lambda &1+6*\lambda &-4*\lambda &\lambda\\0&\cdots&\cdots&\cdots&0 & \lambda & -4*\lambda & 1+5*\lambda & -2*\lambda\\0&\cdots&\cdots&\cdots&\cdots&0 & \lambda & -2*\lambda & 1+\lambda\end{bmatrix} \]

\(y\) - the vector containing the data to be processed

\(\lambda\) - the smoothing parameter.

References

For more details on the method, its advantages, disadvantages and use cases, see the following papers:

• Hodrick, R. J., and Prescott, E. C. (1997). Postwar U.S. Business Cycles: An Empirical Investigation. Journal of Money, Credit, and Banking 29: 1-16. https://doi.org/10.2307/2953682
• Mcelroy, T. (2008). Exact formulas for the Hodrick-Prescott Filter. Econometrics Journal. 11. 209-217. https://doi.org/10.1111/j.1368-423X.2008.00230.x
• BIS (2010). Guidance for national authorities operating the countercyclical capital buffer, Basel Committee on Banking Supervision, ISBN 92-9197-865-5. https://www.bis.org/publ/bcbs187.htm
• Hamilton, J. D. (2017). ‘Why You Should Never Use the Hodrick-Prescott Filter’. Working Paper Series. National Bureau of Economic Research, May 2017. https://doi.org/10.3386/w23429
• Drehmann, M., and Yetman, J. (2018). Why You Should Use the Hodrick-Prescott Filter – at Least to Generate Credit Gaps. BIS Working Paper No. 744. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3275756

hp1 - the one-sided HP Filter

This function applies the one-sided Hodrick-Prescott filter to the selected data, for the given smoothing parameter value.

The one-sided HP filter is obtained by using the Kalman filter to find solutions of the local linear trend model. The approximation produced by the filter requires that a steady-state solution of the Kalman filter can be found. Since, for positive smoothing parameters, this is the case - the method can be used.

In R, simply call the function:

hp1(y, lambda = 1600, x_user, P_user, discard)

The function takes as input:

The function outputs the following series:

hp2 - the Two-Sided HP Filter

This function applies the two-sided Hodrick-Prescott filter to the selected data, for the given smoothing parameter value.

The two-sided HP filter uses a sparse-matrix implementation to enable faster calculation with large datasets.

hp2(y, lambda = 1600)

The function takes as input:

The function outputs the following series:

Studying the GDP cycle with a two-sided HP filter

In this example, we will apply the HP filter to GDP data for the EU from Eurostat. More details about the series can be found in its help file, which you can access by typing: ?GDPEU.

The next code snippet shows the workflow associated with estimating the HP Filter - it involves loading the package, calculating the trend and deriving the cycle.

# R CODE - Smoothing GDP data with the two-sided HP filter
# Load library
library(hpfilter)
# Create the dataframe containing GDP data
y = data.frame(gdp=c(2402903.9, 2416576.3, 2428126.1, 2437407.9, 2443883.8, 2459908.3, 2474557.8, 2487734.5, 2502900.4, 2531872, 2550913.6, 2579582.2, 2598307.2, 2611524.5, 2628335, 2636377, 2662108.5, 2673788.6, 2706474.2, 2739615.4, 2771013.4, 2796237.5, 2810427.4, 2827455.4, 2854217.7, 2860625.6, 2868091.8, 2873472.3, 2879547.6, 2895157.8, 2908087.9, 2918434.9, 2917344.7, 2925034.7, 2945193.3, 2968681, 2985760.1, 3003359.6, 3011430.9, 3024493.3, 3034608.1, 3057625.7, 3082932, 3107995.9, 3134685.3, 3167958.6, 3184861.1, 3215225.8, 3239777.4, 3262175.7, 3280780.3, 3300040, 3318443, 3308233.2, 3285991.3, 3224235.3, 3137585.4, 3134888.1, 3145754.4, 3158823.5, 3173245.5, 3205158.6, 3222254.3, 3239486.8, 3266120.1, 3268551.9, 3275857.1, 3266151.1, 3264311.3, 3257251.5, 3260049.6, 3247444.9, 3243545.5, 3261329.2, 3275659.2, 3288417.7, 3304747.7, 3315578, 3334340.8, 3348942.3, 3372453.9, 3390320.1, 3407279, 3424800.8, 3440244.9, 3451983.1, 3467597.7, 3494562.1, 3519862.5, 3546513.9, 3572479.2, 3598925.7, 3603964.3, 3624883.5, 3630997.8, 3652886.3, 3676573.4, 3689104.4, 3699971.7, 3702073.3))
# Run the two-sided HP filter with the default value of 1600 for lambda
ytrend = hp2(y)
# Calculate cyclical component
ycycle = y - ytrend

Now, we can create graphs to show the results. We first plot the HP filtered trend and the actual series.

# Plot
plot(y$gdp, type="l", col="black", lty=1)
lines(ytrend$gdp, col="#066462")
legend("bottom", horiz=TRUE, cex=0.75, c("y", "ytrend"), lty = 1, col = c("black", "#066462"))

Then, we plot the cyclical component.

plot(ycycle$gdp, type = "l")
abline(h=0)

Synthetic data example for the one-sided HP filter

In this example, we generate random time series data drawn from a normal distribution and apply the one-sided HP filter. After calculating the cyclical component from the trend series generated by the hp1 function, we plot the output alongside the original data.

# Generate the data and plot it
set.seed(10)
y <- as.data.frame(rev(diffinv(rnorm(100)))[1:100])+30
colnames(y) <- "gdp"
plot(y$gdp, type="l")

# Apply the HP filter to the data
ytrend = hp1(y)
ycycle = y - ytrend

# Plot the three resulting series
plot(y$gdp, type="l", col="black", lty=1, ylim=c(-10,30))
lines(ytrend$gdp, col="#066462")
polygon(c(1, seq(ycycle$gdp), length(ycycle$gdp)), c(0, ycycle$gdp, 0), col = "#E0F2F1")
legend("bottom", horiz=TRUE, cex=0.75, c("y", "ytrend", "ycycle"), lty = 1,
       col = c("black", "#066462", "#75bfbd"))