# coding: utf-8
# DO NOT EDIT
# Autogenerated from the notebook statespace_structural_harvey_jaeger.ipynb.
# Edit the notebook and then sync the output with this file.
#
# flake8: noqa
# DO NOT EDIT
# # Detrending, Stylized Facts and the Business Cycle
#
# In an influential article, Harvey and Jaeger (1993) described the use of
# unobserved components models (also known as "structural time series
# models") to derive stylized facts of the business cycle.
#
# Their paper begins:
#
# "Establishing the 'stylized facts' associated with a set of time
# series is widely considered a crucial step
# in macroeconomic research ... For such facts to be useful they
# should (1) be consistent with the stochastic
# properties of the data and (2) present meaningful information."
#
# In particular, they make the argument that these goals are often better
# met using the unobserved components approach rather than the popular
# Hodrick-Prescott filter or Box-Jenkins ARIMA modeling techniques.
#
# statsmodels has the ability to perform all three types of analysis, and
# below we follow the steps of their paper, using a slightly updated
# dataset.
import numpy as np
import pandas as pd
import statsmodels.api as sm
import matplotlib.pyplot as plt
from IPython.display import display, Latex
# ## Unobserved Components
#
# The unobserved components model available in statsmodels can be written
# as:
#
# $$
# y_t = \underbrace{\mu_{t}}_{\text{trend}} +
# \underbrace{\gamma_{t}}_{\text{seasonal}} +
# \underbrace{c_{t}}_{\text{cycle}} + \sum_{j=1}^k \underbrace{\beta_j
# x_{jt}}_{\text{explanatory}} +
# \underbrace{\varepsilon_t}_{\text{irregular}}
# $$
#
# see Durbin and Koopman 2012, Chapter 3 for notation and additional
# details. Notice that different specifications for the different individual
# components can support a wide range of models. The specific models
# considered in the paper and below are specializations of this general
# equation.
#
# ### Trend
#
# The trend component is a dynamic extension of a regression model that
# includes an intercept and linear time-trend.
#
# $$
# \begin{align}
# \underbrace{\mu_{t+1}}_{\text{level}} & = \mu_t + \nu_t + \eta_{t+1}
# \qquad & \eta_{t+1} \sim N(0, \sigma_\eta^2) \\\\
# \underbrace{\nu_{t+1}}_{\text{trend}} & = \nu_t + \zeta_{t+1} &
# \zeta_{t+1} \sim N(0, \sigma_\zeta^2) \\
# \end{align}
# $$
#
# where the level is a generalization of the intercept term that can
# dynamically vary across time, and the trend is a generalization of the
# time-trend such that the slope can dynamically vary across time.
#
# For both elements (level and trend), we can consider models in which:
#
# - The element is included vs excluded (if the trend is included, there
# must also be a level included).
# - The element is deterministic vs stochastic (i.e. whether or not the
# variance on the error term is confined to be zero or not)
#
# The only additional parameters to be estimated via MLE are the variances
# of any included stochastic components.
#
# This leads to the following specifications:
#
# | |
# Level | Trend | Stochastic Level | Stochastic Trend |
# |----------------------------------------------------------------------|
# -------|-------|------------------|------------------|
# | Constant |
# ✓ | | | |
# | Local Level
(random walk) |
# ✓ | | ✓ | |
# | Deterministic trend |
# ✓ | ✓ | | |
# | Local level with deterministic trend
(random walk with drift) |
# ✓ | ✓ | ✓ | |
# | Local linear trend |
# ✓ | ✓ | ✓ | ✓ |
# | Smooth trend
(integrated random walk) |
# ✓ | ✓ | | ✓ |
#
# ### Seasonal
#
# The seasonal component is written as:
#
# $$
# \gamma_t = - \sum_{j=1}^{s-1} \gamma_{t+1-j} + \omega_t \qquad \omega_t
# \sim N(0, \sigma_\omega^2)
# $$
#
# The periodicity (number of seasons) is `s`, and the defining character
# is that (without the error term), the seasonal components sum to zero
# across one complete cycle. The inclusion of an error term allows the
# seasonal effects to vary over time.
#
# The variants of this model are:
#
# - The periodicity `s`
# - Whether or not to make the seasonal effects stochastic.
#
# If the seasonal effect is stochastic, then there is one additional
# parameter to estimate via MLE (the variance of the error term).
#
# ### Cycle
#
# The cyclical component is intended to capture cyclical effects at time
# frames much longer than captured by the seasonal component. For example,
# in economics the cyclical term is often intended to capture the business
# cycle, and is then expected to have a period between "1.5 and 12 years"
# (see Durbin and Koopman).
#
# The cycle is written as:
#
# $$
# \begin{align}
# c_{t+1} & = c_t \cos \lambda_c + c_t^* \sin \lambda_c + \tilde \omega_t
# \qquad & \tilde \omega_t \sim N(0, \sigma_{\tilde \omega}^2) \\\\
# c_{t+1}^* & = -c_t \sin \lambda_c + c_t^* \cos \lambda_c + \tilde
# \omega_t^* & \tilde \omega_t^* \sim N(0, \sigma_{\tilde \omega}^2)
# \end{align}
# $$
#
# The parameter $\lambda_c$ (the frequency of the cycle) is an additional
# parameter to be estimated by MLE. If the seasonal effect is stochastic,
# then there is one another parameter to estimate (the variance of the error
# term - note that both of the error terms here share the same variance, but
# are assumed to have independent draws).
#
# ### Irregular
#
# The irregular component is assumed to be a white noise error term. Its
# variance is a parameter to be estimated by MLE; i.e.
#
# $$
# \varepsilon_t \sim N(0, \sigma_\varepsilon^2)
# $$
#
# In some cases, we may want to generalize the irregular component to
# allow for autoregressive effects:
#
# $$
# \varepsilon_t = \rho(L) \varepsilon_{t-1} + \epsilon_t, \qquad
# \epsilon_t \sim N(0, \sigma_\epsilon^2)
# $$
#
# In this case, the autoregressive parameters would also be estimated via
# MLE.
#
# ### Regression effects
#
# We may want to allow for explanatory variables by including additional
# terms
#
# $$
# \sum_{j=1}^k \beta_j x_{jt}
# $$
#
# or for intervention effects by including
#
# $$
# \begin{align}
# \delta w_t \qquad \text{where} \qquad w_t & = 0, \qquad t < \tau, \\\\
# & = 1, \qquad t \ge \tau
# \end{align}
# $$
#
# These additional parameters could be estimated via MLE or by including
# them as components of the state space formulation.
#
# ## Data
#
# Following Harvey and Jaeger, we will consider the following time series:
#
# - US real GNP, "output",
# ([GNPC96](https://research.stlouisfed.org/fred2/series/GNPC96))
# - US GNP implicit price deflator, "prices",
# ([GNPDEF](https://research.stlouisfed.org/fred2/series/GNPDEF))
# - US monetary base, "money",
# ([AMBSL](https://research.stlouisfed.org/fred2/series/AMBSL))
#
# The time frame in the original paper varied across series, but was
# broadly 1954-1989. Below we use data from the period 1948-2008 for all
# series. Although the unobserved components approach allows isolating a
# seasonal component within the model, the series considered in the paper,
# and here, are already seasonally adjusted.
#
# All data series considered here are taken from [Federal Reserve Economic
# Data (FRED)](https://research.stlouisfed.org/fred2/). Conveniently, the
# Python library [Pandas](http://pandas.pydata.org/) has the ability to
# download data from FRED directly.
# Datasets
from pandas_datareader.data import DataReader
# Get the raw data
start = '1948-01'
end = '2008-01'
us_gnp = DataReader('GNPC96', 'fred', start=start, end=end)
us_gnp_deflator = DataReader('GNPDEF', 'fred', start=start, end=end)
us_monetary_base = DataReader(
'AMBSL', 'fred', start=start, end=end).resample('QS').mean()
recessions = DataReader(
'USRECQ', 'fred', start=start, end=end).resample('QS').last().values[:, 0]
# Construct the dataframe
dta = pd.concat(
map(np.log, (us_gnp, us_gnp_deflator, us_monetary_base)), axis=1)
dta.columns = ['US GNP', 'US Prices', 'US monetary base']
dates = dta.index._mpl_repr()
# To get a sense of these three variables over the timeframe, we can plot
# them:
# Plot the data
ax = dta.plot(figsize=(13, 3))
ylim = ax.get_ylim()
ax.xaxis.grid()
ax.fill_between(
dates,
ylim[0] + 1e-5,
ylim[1] - 1e-5,
recessions,
facecolor='k',
alpha=0.1)
# ## Model
#
# Since the data is already seasonally adjusted and there are no obvious
# explanatory variables, the generic model considered is:
#
# $$
# y_t = \underbrace{\mu_{t}}_{\text{trend}} +
# \underbrace{c_{t}}_{\text{cycle}} +
# \underbrace{\varepsilon_t}_{\text{irregular}}
# $$
#
# The irregular will be assumed to be white noise, and the cycle will be
# stochastic and damped. The final modeling choice is the specification to
# use for the trend component. Harvey and Jaeger consider two models:
#
# 1. Local linear trend (the "unrestricted" model)
# 2. Smooth trend (the "restricted" model, since we are forcing
# $\sigma_\eta = 0$)
#
# Below, we construct `kwargs` dictionaries for each of these model types.
# Notice that rather that there are two ways to specify the models. One way
# is to specify components directly, as in the table above. The other way is
# to use string names which map to various specifications.
# Model specifications
# Unrestricted model, using string specification
unrestricted_model = {
'level': 'local linear trend',
'cycle': True,
'damped_cycle': True,
'stochastic_cycle': True
}
# Unrestricted model, setting components directly
# This is an equivalent, but less convenient, way to specify a
# local linear trend model with a stochastic damped cycle:
# unrestricted_model = {
# 'irregular': True, 'level': True, 'stochastic_level': True, 'trend':
# True, 'stochastic_trend': True,
# 'cycle': True, 'damped_cycle': True, 'stochastic_cycle': True
# }
# The restricted model forces a smooth trend
restricted_model = {
'level': 'smooth trend',
'cycle': True,
'damped_cycle': True,
'stochastic_cycle': True
}
# Restricted model, setting components directly
# This is an equivalent, but less convenient, way to specify a
# smooth trend model with a stochastic damped cycle. Notice
# that the difference from the local linear trend model is that
# `stochastic_level=False` here.
# unrestricted_model = {
# 'irregular': True, 'level': True, 'stochastic_level': False,
# 'trend': True, 'stochastic_trend': True,
# 'cycle': True, 'damped_cycle': True, 'stochastic_cycle': True
# }
# We now fit the following models:
#
# 1. Output, unrestricted model
# 2. Prices, unrestricted model
# 3. Prices, restricted model
# 4. Money, unrestricted model
# 5. Money, restricted model
# Output
output_mod = sm.tsa.UnobservedComponents(dta['US GNP'], **unrestricted_model)
output_res = output_mod.fit(method='powell', disp=False)
# Prices
prices_mod = sm.tsa.UnobservedComponents(dta['US Prices'],
**unrestricted_model)
prices_res = prices_mod.fit(method='powell', disp=False)
prices_restricted_mod = sm.tsa.UnobservedComponents(dta['US Prices'],
**restricted_model)
prices_restricted_res = prices_restricted_mod.fit(method='powell', disp=False)
# Money
money_mod = sm.tsa.UnobservedComponents(dta['US monetary base'],
**unrestricted_model)
money_res = money_mod.fit(method='powell', disp=False)
money_restricted_mod = sm.tsa.UnobservedComponents(dta['US monetary base'],
**restricted_model)
money_restricted_res = money_restricted_mod.fit(method='powell', disp=False)
# Once we have fit these models, there are a variety of ways to display
# the information. Looking at the model of US GNP, we can summarize the fit
# of the model using the `summary` method on the fit object.
print(output_res.summary())
# For unobserved components models, and in particular when exploring
# stylized facts in line with point (2) from the introduction, it is often
# more instructive to plot the estimated unobserved components (e.g. the
# level, trend, and cycle) themselves to see if they provide a meaningful
# description of the data.
#
# The `plot_components` method of the fit object can be used to show plots
# and confidence intervals of each of the estimated states, as well as a
# plot of the observed data versus the one-step-ahead predictions of the
# model to assess fit.
fig = output_res.plot_components(
legend_loc='lower right', figsize=(15, 9))
# Finally, Harvey and Jaeger summarize the models in another way to
# highlight the relative importances of the trend and cyclical components;
# below we replicate their Table I. The values we find are broadly
# consistent with, but different in the particulars from, the values from
# their table.
# Create Table I
table_i = np.zeros((5, 6))
start = dta.index[0]
end = dta.index[-1]
time_range = '%d:%d-%d:%d' % (start.year, start.quarter, end.year, end.quarter)
models = [
('US GNP', time_range, 'None'),
('US Prices', time_range, 'None'),
('US Prices', time_range, r'$\sigma_\eta^2 = 0$'),
('US monetary base', time_range, 'None'),
('US monetary base', time_range, r'$\sigma_\eta^2 = 0$'),
]
index = pd.MultiIndex.from_tuples(
models, names=['Series', 'Time range', 'Restrictions'])
parameter_symbols = [
r'$\sigma_\zeta^2$',
r'$\sigma_\eta^2$',
r'$\sigma_\kappa^2$',
r'$\rho$',
r'$2 \pi / \lambda_c$',
r'$\sigma_\varepsilon^2$',
]
i = 0
for res in (output_res, prices_res, prices_restricted_res, money_res,
money_restricted_res):
if res.model.stochastic_level:
(sigma_irregular, sigma_level, sigma_trend, sigma_cycle,
frequency_cycle, damping_cycle) = res.params
else:
(sigma_irregular, sigma_level, sigma_cycle, frequency_cycle,
damping_cycle) = res.params
sigma_trend = np.nan
period_cycle = 2 * np.pi / frequency_cycle
table_i[i, :] = [
sigma_level * 1e7, sigma_trend * 1e7, sigma_cycle * 1e7, damping_cycle,
period_cycle, sigma_irregular * 1e7
]
i += 1
pd.set_option('float_format',
lambda x: '%.4g' % np.round(x, 2) if not np.isnan(x) else '-')
table_i = pd.DataFrame(table_i, index=index, columns=parameter_symbols)
table_i