2026-11-05
\[ \newcommand{\vy}{\mathbf{y}} \newcommand{\vx}{\mathbf{x}} \newcommand{\vb}{\mathbf{b}} \newcommand{\ve}{\mathbf{e}} \newcommand{\vone}{\mathbf{1}} \newcommand{\vbeta}{\boldsymbol{\beta}} \newcommand{\vmu}{\boldsymbol{\mu}} \newcommand{\mX}{\mathbf{X}} \newcommand{\mZ}{\mathbf{Z}} \newcommand{\mV}{\mathbf{V}} \newcommand{\mI}{\mathbf{I}} \newcommand{\mJ}{\mathbf{J}} \newcommand{\mSigma}{\boldsymbol{\Sigma}} \newcommand{\vg}{\mathbf{g}} \newcommand{\vc}{\mathbf{c}} \newcommand{\vz}{\mathbf{z}} \newcommand{\vlambda}{\boldsymbol{\lambda}} \newcommand{\mA}{\mathbf{A}} \newcommand{\mB}{\mathbf{B}} \newcommand{\mK}{\mathbf{K}} \newcommand{\mD}{\mathbf{D}} \newcommand{\mL}{\mathbf{L}} \newcommand{\mQ}{\mathbf{Q}} \newcommand{\E}{\mathrm{E}} \newcommand{\Var}{\mathrm{Var}} \newcommand{\Cov}{\mathrm{Cov}} \newcommand{\tr}{\operatorname{tr}} \]
mhpfilter splits a macroeconomic time series into a trend and a cycle. This vignette takes the quarterly real GDP of the United States from the raw series to the three filters of the package, their comparison, many series at once and a plain-language reading of the results. The later vignettes give the theory (vignette 2), the two papers behind the package (vignettes 3 and 4), the simulation study (vignette 5), the application (vignette 6) and the checks of the code (vignette 7).
us_macro holds quarterly real GDP, consumption and investment of the United States from 1959 to 2009 (FRED). As a ts the series keeps its dates, and every result of the package keeps them too.
data(us_macro)
head(us_macro) year quarter gdp consumption investment
1 1959 1 2710.349 1707.4 286.898
2 1959 2 2778.801 1733.7 310.859
3 1959 3 2775.488 1751.8 289.226
4 1959 4 2785.204 1753.7 299.356
5 1960 1 2847.699 1770.5 331.722
6 1960 2 2834.390 1792.9 298.152gdp <- ts(data = us_macro$gdp, start = c(1959, 1), frequency = 4)The filter of Hodrick and Prescott (1997) fits a smooth trend \(\vg\) to the series \(\vx\) by minimising \[ \sum_{t=1}^{T}(x_t - g_t)^2 + \lambda\sum_{t=2}^{T-1}\left[(g_{t+1} - g_t) - (g_t - g_{t-1})\right]^2 , \] and the cycle is \(\vc = \vx - \vg\). The smoothing parameter \(\lambda\) is conventionally 1600 for quarterly data. On the logarithm of a series the cycle is close to the percentage deviation from the trend.
hp <-
hp_filter(
x = log(gdp)
, lambda = 1600
, as_dt = TRUE
, log_scale = FALSE
)
hpWith as_dt = TRUE, the default kept from version 0.1.0, the result is a data.table; as_dt = FALSE gives a list with methods for printing and plotting.
The modified HP filter of McDermott (1997), assessed by Choudhary, Hanif and Iqbal (2014), lets the data choose \(\lambda\): it tries every \(\lambda = 1, 2, \dots\) and keeps the one that minimises a generalised cross-validation criterion.
mhp <-
mhp_filter(
x = log(gdp)
, max_lambda = 100000
, as_dt = FALSE
, log_scale = FALSE
, criterion = "approximate"
)
mhpModified HP Filter Result
==============================
Observations: 203
Optimal lambda: 683
GCV: 2.82e-04
Cycle Statistics:
Mean: -1.634e-13
SD: 0.01333
AR1: 0.828
Range: 0.07802
Sources: Hodrick and Prescott (1997); McDermott (1997); Coe and McDermott
(1997); Choudhary et al. (2014); Choudhary et al. (2014, MATLAB code). Full
references: mhp_cite(x).plot(x = mhp)
plot(x = mhp, which = "gcv")
The HP filter has too little information at the ends of a series, so its trend follows the last observations too closely and the end of the cycle is biased towards zero, just where a policy maker needs it most. The fully modified HP filter of Hanif, Iqbal and Choudhary (2017) raises \(\lambda\) at the first and last observations, with weights chosen to make the filter at the ends behave like the filter in the middle, and chooses \(\lambda\) from the data as the modified HP filter does. The published code filters the logarithm of the series and returns the trend and the cycle in levels; log_scale = TRUE does the same.
fm <-
fmhp_filter(
x = gdp
, max_lambda = 3000
, k_linear = 15
, k_quadratic = NULL
, alpha_linear = seq(from = 0, to = 2000, by = 10)
, alpha_quadratic = seq(from = 0, to = 2000, by = 5)
, schemes = c("linear", "quadratic")
, form = "rows"
, lambda = NULL
, log_scale = TRUE
, as_dt = FALSE
)
fmFully modified HP Filter Result
==============================
Observations: 203
Fully modified lambda: 1448 (modified HP lambda: 683 )
End-point weights: quadratic, alpha = 15 on 22 end values; loss = 0.6163
GCV: 3.336e-04
Filtered on the log scale; cycle = series - trend in levels
Cycle Statistics:
Mean: -2.603
SD: 120.4
AR1: 0.9522
Range: 900
Sources: Hanif et al. (2017); Hanif (2017); Bloechl (2014); McDermott (1997);
Hodrick and Prescott (1997). Full references: mhp_cite(x).fm$schemes scheme k alpha loss lambda gcv
<char> <int> <num> <num> <int> <num>
1: linear 15 240 0.8548936 1169 0.0003168500
2: quadratic 22 15 0.6163442 1448 0.0003335623plot(x = fm, which = "lambda")
The smoothing parameter rises from 1448 in the middle to 8708 at the first and last observation (quadratic weights on 22 values at each end).
fmhp_compare() filters the series with the three filters and compares the cycles and their end-point loss, the measure of end-point bias used by Hanif et al. (2017): zero for a filter that treats the ends like the middle.
cmp <-
fmhp_compare(
x = gdp
, frequency = "quarterly"
, max_lambda = 3000
, log_scale = TRUE
, end_share = 0.1
)
cmp method lambda cycle_sd cycle_mean ar1 cycle_range
<char> <num> <num> <num> <num> <num>
1: HP 1600 0.01543904 4.942713e-12 0.8668207 0.08590516
2: Modified HP 683 0.01333176 -1.634336e-13 0.8280266 0.07801575
3: Fully modified HP 1448 0.01618038 -4.145448e-05 0.8815215 0.08713978
gcv loss sd_ends sd_middle
<num> <num> <num> <num>
1: NA 1.7617778 0.01458541 0.01568213
2: 0.0002819926 1.7652928 0.01202396 0.01366805
3: 0.0003335623 0.7898641 0.01930994 0.01538061The fully modified HP filter cuts the loss of the HP filter by more than half. Its cycle is also more variable at the ends than in the middle, unlike the cycles of the other two filters: the end-point bias of the HP filter shows as cycles that are too small at the ends.
plot_comparison(
x = log(gdp)
, frequency = "quarterly"
, max_lambda = 100000
, show_cycle = TRUE
, fmhp = TRUE
)
mhp_batch() and fmhp_batch() filter every column of a matrix; series of different lengths can share it, padded with missing values at the start or the end.
X <- as.matrix(us_macro[, c("gdp", "consumption", "investment")])
fb <-
fmhp_batch(
X = X
, max_lambda = 3000
, k_linear = 15
, k_quadratic = NULL
, alpha_linear = seq(from = 0, to = 2000, by = 10)
, alpha_quadratic = seq(from = 0, to = 2000, by = 5)
, schemes = c("linear", "quadratic")
, form = "rows"
, lambda = NULL
, log_scale = TRUE
)
attr(x = fb, which = "lambdas") series lambda gcv lambda_mhp scheme alpha k loss
<char> <int> <num> <int> <char> <num> <int> <num>
1: gdp 1448 0.0003335623 683 quadratic 15 22 0.6163442
2: consumption 1261 0.0002248243 534 quadratic 15 22 0.5850792
3: investment 2961 0.0073465762 1009 quadratic 20 22 0.6728679plot_batch(
x = fb
, show = "cycle"
, facet = TRUE
, highlight = NULL
)
mhp_interpret() reads a result in plain language, from its numbers, and ends with the sources of the method.
mhp_interpret(
x = fm
, top = 3
)Fully modified HP filter
The fully modified HP filter chose lambda = 1,448 (the modified HP filter
alone: 683) and quadratic end-point weights on 22 values at each end, which
cut the end-point loss of the HP filter with the same lambda by 65%.
Findings:
- Cycle (log scale, about the share of the trend): standard deviation 0.0162,
first-order autocorrelation 0.88, range 0.0871.
- Largest positive cycle values (booms) at 2000Q2, 1978Q4, 1999Q4; largest
negative ones (recessions) at 2009Q2, 2009Q3, 2009Q1.
- Linear weights: loss 0.855 (alpha = 240); quadratic weights: loss 0.616
(alpha = 15); the quadratic scheme was kept.
Recommendations:
- Compare the filters on this series with fmhp_compare() (moments of the
cycles and end-point loss).
Cautions:
- The last values of the cycle are revised as new observations arrive: read
them with care.
Sources:
- Hanif, M. N., Iqbal, J. and Choudhary, M. A. (2017). Fully modified HP
filter. SBP Working Paper Series, No. 88, State Bank of Pakistan.
- Hanif, M. N. (2017). Fully modified HP filter function (MATLAB code).
MATLAB Central File Exchange, 63198.
- Bloechl, A. (2014). Reducing the excess variability of the Hodrick-Prescott
filter by flexible penalization. Discussion Paper 2014-1, University of
Munich.
- McDermott, C. J. (1997). An automatic method for choosing the smoothing
parameter in the HP filter. Unpublished, International Monetary Fund,
Washington DC.
- Hodrick, R. J. and Prescott, E. C. (1997). Postwar U.S. business cycles: an
empirical investigation. Journal of Money, Credit and Banking, 29(1), 1-16.
doi:10.2307/2953682mhp_app().Choudhary, M. A., Hanif, M. N. and Iqbal, J. (2014). On smoothing macroeconomic time series using the modified HP filter. Applied Economics, 46(19), 2205–2214.
Hanif, M. N., Iqbal, J. and Choudhary, M. A. (2017). Fully modified HP filter. SBP Working Paper Series, No. 88, State Bank of Pakistan.
Hodrick, R. J. and Prescott, E. C. (1997). Postwar U.S. business cycles: an empirical investigation. Journal of Money, Credit and Banking, 29(1), 1–16.
McDermott, C. J. (1997). An automatic method for choosing the smoothing parameter in the HP filter. Unpublished, International Monetary Fund.