# Source code for statsmodels.tsa.filters.hp_filter

```from __future__ import absolute_import

from scipy import sparse
from scipy.sparse.linalg import spsolve
import numpy as np
from ._utils import _maybe_get_pandas_wrapper

[docs]def hpfilter(X, lamb=1600):
"""
Hodrick-Prescott filter

Parameters
----------
X : array-like
The 1d ndarray timeseries to filter of length (nobs,) or (nobs,1)
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 : array
The estimated cycle in the data given lamb.
trend : array
The estimated trend in the data given lamb.

Examples
---------
>>> import statsmodels.api as sm
>>> import pandas as pd
>>> dta = sm.datasets.macrodata.load_pandas().data
>>> index = pd.DatetimeIndex(start='1959Q1', end='2009Q4', 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

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

--------
statsmodels.tsa.filters.bk_filter.bkfilter
statsmodels.tsa.filters.cf_filter.cffilter
statsmodels.tsa.seasonal.seasonal_decompose

References
----------
Hodrick, R.J, and E. C. Prescott. 1980. "Postwar U.S. Business Cycles: An
Empricial 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.
"""
_pandas_wrapper = _maybe_get_pandas_wrapper(X)
X = np.asarray(X, float)
if X.ndim > 1:
X = X.squeeze()
nobs = len(X)
I = sparse.eye(nobs,nobs)
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
if _pandas_wrapper is not None:
return _pandas_wrapper(cycle), _pandas_wrapper(trend)
return cycle, trend
```