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


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

    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')