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#!/usr/bin/env python
# coding: utf-8
# DO NOT EDIT
# Autogenerated from the notebook statespace_cycles.ipynb.
# Edit the notebook and then sync the output with this file.
#
# flake8: noqa
# DO NOT EDIT
# # Trends and cycles in unemployment
#
# Here we consider three methods for separating a trend and cycle in
# economic data. Supposing we have a time series $y_t$, the basic idea is to
# decompose it into these two components:
#
# $$
# y_t = \mu_t + \eta_t
# $$
#
# where $\mu_t$ represents the trend or level and $\eta_t$ represents the
# cyclical component. In this case, we consider a *stochastic* trend, so
# that $\mu_t$ is a random variable and not a deterministic function of
# time. Two of methods fall under the heading of "unobserved components"
# models, and the third is the popular Hodrick-Prescott (HP) filter.
# Consistent with e.g. Harvey and Jaeger (1993), we find that these models
# all produce similar decompositions.
#
# This notebook demonstrates applying these models to separate trend from
# cycle in the U.S. unemployment rate.
import numpy as np
import pandas as pd
import statsmodels.api as sm
import matplotlib.pyplot as plt
from pandas_datareader.data import DataReader
endog = DataReader('UNRATE', 'fred', start='1954-01-01')
endog.index.freq = endog.index.inferred_freq
# ### Hodrick-Prescott (HP) filter
#
# The first method is the Hodrick-Prescott filter, which can be applied to
# a data series in a very straightforward method. Here we specify the
# parameter $\lambda=129600$ because the unemployment rate is observed
# monthly.
hp_cycle, hp_trend = sm.tsa.filters.hpfilter(endog, lamb=129600)
# ### Unobserved components and ARIMA model (UC-ARIMA)
#
# The next method is an unobserved components model, where the trend is
# modeled as a random walk and the cycle is modeled with an ARIMA model - in
# particular, here we use an AR(4) model. The process for the time series
# can be written as:
#
# $$
# \begin{align}
# y_t & = \mu_t + \eta_t \\
# \mu_{t+1} & = \mu_t + \epsilon_{t+1} \\
# \phi(L) \eta_t & = \nu_t
# \end{align}
# $$
#
# where $\phi(L)$ is the AR(4) lag polynomial and $\epsilon_t$ and $\nu_t$
# are white noise.
mod_ucarima = sm.tsa.UnobservedComponents(endog, 'rwalk', autoregressive=4)
# Here the powell method is used, since it achieves a
# higher loglikelihood than the default L-BFGS method
res_ucarima = mod_ucarima.fit(method='powell', disp=False)
print(res_ucarima.summary())
# ### Unobserved components with stochastic cycle (UC)
#
# The final method is also an unobserved components model, but where the
# cycle is modeled explicitly.
#
# $$
# \begin{align}
# y_t & = \mu_t + \eta_t \\
# \mu_{t+1} & = \mu_t + \epsilon_{t+1} \\
# \eta_{t+1} & = \eta_t \cos \lambda_\eta + \eta_t^* \sin \lambda_\eta +
# \tilde \omega_t \qquad & \tilde \omega_t \sim N(0, \sigma_{\tilde
# \omega}^2) \\
# \eta_{t+1}^* & = -\eta_t \sin \lambda_\eta + \eta_t^* \cos \lambda_\eta
# + \tilde \omega_t^* & \tilde \omega_t^* \sim N(0, \sigma_{\tilde
# \omega}^2)
# \end{align}
# $$
mod_uc = sm.tsa.UnobservedComponents(
endog,
'rwalk',
cycle=True,
stochastic_cycle=True,
damped_cycle=True,
)
# Here the powell method gets close to the optimum
res_uc = mod_uc.fit(method='powell', disp=False)
# but to get to the highest loglikelihood we do a
# second round using the L-BFGS method.
res_uc = mod_uc.fit(res_uc.params, disp=False)
print(res_uc.summary())
# ### Graphical comparison
#
# The output of each of these models is an estimate of the trend component
# $\mu_t$ and an estimate of the cyclical component $\eta_t$. Qualitatively
# the estimates of trend and cycle are very similar, although the trend
# component from the HP filter is somewhat more variable than those from the
# unobserved components models. This means that relatively mode of the
# movement in the unemployment rate is attributed to changes in the
# underlying trend rather than to temporary cyclical movements.
fig, axes = plt.subplots(2, figsize=(13, 5))
axes[0].set(title='Level/trend component')
axes[0].plot(endog.index, res_uc.level.smoothed, label='UC')
axes[0].plot(endog.index, res_ucarima.level.smoothed, label='UC-ARIMA(2,0)')
axes[0].plot(hp_trend, label='HP Filter')
axes[0].legend(loc='upper left')
axes[0].grid()
axes[1].set(title='Cycle component')
axes[1].plot(endog.index, res_uc.cycle.smoothed, label='UC')
axes[1].plot(endog.index,
res_ucarima.autoregressive.smoothed,
label='UC-ARIMA(2,0)')
axes[1].plot(hp_cycle, label='HP Filter')
axes[1].legend(loc='upper left')
axes[1].grid()
fig.tight_layout()
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