# Source code for statsmodels.tsa.filters.hp_filter

import numpy as np
from scipy import sparse
from scipy.sparse.linalg import spsolve
from statsmodels.tools.validation import array_like, PandasWrapper

[docs] def hpfilter(x, lamb=1600): """ Hodrick-Prescott filter. Parameters ---------- x : array_like The time series to filter, 1-d. lamb : float The Hodrick-Prescott smoothing parameter. A value of 1600 is suggested for quarterly data. Ravn and Uhlig suggest using a value of 6.25 (1600/4**4) for annual data and 129600 (1600*3**4) for monthly data. Returns ------- cycle : ndarray The estimated cycle in the data given lamb. trend : ndarray The estimated trend in the data given lamb. See Also -------- statsmodels.tsa.filters.bk_filter.bkfilter Baxter-King filter. statsmodels.tsa.filters.cf_filter.cffilter The Christiano Fitzgerald asymmetric, random walk filter. statsmodels.tsa.seasonal.seasonal_decompose Decompose a time series using moving averages. statsmodels.tsa.seasonal.STL Season-Trend decomposition using LOESS. Notes ----- The HP filter removes a smooth trend, `T`, from the data `x`. by solving min sum((x[t] - T[t])**2 + lamb*((T[t+1] - T[t]) - (T[t] - T[t-1]))**2) T t Here we implemented the HP filter as a ridge-regression rule using scipy.sparse. In this sense, the solution can be written as T = inv(I + lamb*K'K)x where I is a nobs x nobs identity matrix, and K is a (nobs-2) x nobs matrix such that K[i,j] = 1 if i == j or i == j + 2 K[i,j] = -2 if i == j + 1 K[i,j] = 0 otherwise See the notebook `Time Series Filters <../examples/notebooks/generated/tsa_filters.html>`__ for an overview. References ---------- Hodrick, R.J, and E. C. Prescott. 1980. "Postwar U.S. Business Cycles: An Empirical Investigation." `Carnegie Mellon University discussion paper no. 451`. Ravn, M.O and H. Uhlig. 2002. "Notes On Adjusted the Hodrick-Prescott Filter for the Frequency of Observations." `The Review of Economics and Statistics`, 84(2), 371-80. Examples -------- >>> import statsmodels.api as sm >>> import pandas as pd >>> dta = sm.datasets.macrodata.load_pandas().data >>> index = pd.period_range('1959Q1', '2009Q3', freq='Q') >>> dta.set_index(index, inplace=True) >>> cycle, trend = sm.tsa.filters.hpfilter(dta.realgdp, 1600) >>> gdp_decomp = dta[['realgdp']] >>> gdp_decomp["cycle"] = cycle >>> gdp_decomp["trend"] = trend >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> gdp_decomp[["realgdp", "trend"]]["2000-03-31":].plot(ax=ax, ... fontsize=16) >>> plt.show() .. plot:: plots/hpf_plot.py """ pw = PandasWrapper(x) x = array_like(x, 'x', ndim=1) nobs = len(x) I = sparse.eye(nobs, nobs) # noqa:E741 offsets = np.array([0, 1, 2]) data = np.repeat([[1.], [-2.], [1.]], nobs, axis=1) K = sparse.dia_matrix((data, offsets), shape=(nobs - 2, nobs)) use_umfpack = True trend = spsolve(I+lamb*K.T.dot(K), x, use_umfpack=use_umfpack) cycle = x - trend return pw.wrap(cycle, append='cycle'), pw.wrap(trend, append='trend')

Last update: Apr 12, 2024