#!/usr/bin/env python
# coding: utf-8

# DO NOT EDIT
# Autogenerated from the notebook tsa_filters.ipynb.
# Edit the notebook and then sync the output with this file.
#
# flake8: noqa
# DO NOT EDIT

# # Time Series Filters

import pandas as pd
import matplotlib.pyplot as plt

import statsmodels.api as sm

dta = sm.datasets.macrodata.load_pandas().data

index = pd.Index(sm.tsa.datetools.dates_from_range("1959Q1", "2009Q3"))
print(index)

dta.index = index
del dta["year"]
del dta["quarter"]

print(sm.datasets.macrodata.NOTE)

print(dta.head(10))

fig = plt.figure(figsize=(12, 8))
ax = fig.add_subplot(111)
dta.realgdp.plot(ax=ax)
legend = ax.legend(loc="upper left")
legend.prop.set_size(20)

# ### Hodrick-Prescott Filter

# The Hodrick-Prescott filter separates a time-series $y_t$ into a trend
# $\tau_t$ and a cyclical component $\zeta_t$
#
# $$y_t = \tau_t + \zeta_t$$
#
# The components are determined by minimizing the following quadratic loss
# function
#
# $$\min_{\\{ \tau_{t}\\} }\sum_{t}^{T}\zeta_{t}^{2}+\lambda\sum_{t=1}^{T}
# \left[\left(\tau_{t}-\tau_{t-1}\right)-\left(\tau_{t-1}-\tau_{t-
# 2}\right)\right]^{2}$$

gdp_cycle, gdp_trend = sm.tsa.filters.hpfilter(dta.realgdp)

gdp_decomp = dta[["realgdp"]].copy()
gdp_decomp["cycle"] = gdp_cycle
gdp_decomp["trend"] = gdp_trend

fig = plt.figure(figsize=(12, 8))
ax = fig.add_subplot(111)
gdp_decomp[["realgdp", "trend"]]["2000-03-31":].plot(ax=ax, fontsize=16)
legend = ax.get_legend()
legend.prop.set_size(20)

# ### Baxter-King approximate band-pass filter: Inflation and Unemployment

# #### Explore the hypothesis that inflation and unemployment are counter-
# cyclical.

# The Baxter-King filter is intended to explicitly deal with the
# periodicity of the business cycle. By applying their band-pass filter to a
# series, they produce a new series that does not contain fluctuations at
# higher or lower than those of the business cycle. Specifically, the BK
# filter takes the form of a symmetric moving average
#
# $$y_{t}^{*}=\sum_{k=-K}^{k=K}a_ky_{t-k}$$
#
# where $a_{-k}=a_k$ and $\sum_{k=-k}^{K}a_k=0$ to eliminate any trend in
# the series and render it stationary if the series is I(1) or I(2).
#
# For completeness, the filter weights are determined as follows
#
# $$a_{j} = B_{j}+\theta\text{ for }j=0,\pm1,\pm2,\dots,\pm K$$
#
# $$B_{0} = \frac{\left(\omega_{2}-\omega_{1}\right)}{\pi}$$
# $$B_{j} = \frac{1}{\pi j}\left(\sin\left(\omega_{2}j\right)-
# \sin\left(\omega_{1}j\right)\right)\text{ for }j=0,\pm1,\pm2,\dots,\pm K$$
#
# where $\theta$ is a normalizing constant such that the weights sum to
# zero.
#
# $$\theta=\frac{-\sum_{j=-K^{K}b_{j}}}{2K+1}$$
#
# $$\omega_{1}=\frac{2\pi}{P_{H}}$$
#
# $$\omega_{2}=\frac{2\pi}{P_{L}}$$
#
# $P_L$ and $P_H$ are the periodicity of the low and high cut-off
# frequencies. Following Burns and Mitchell's work on US business cycles
# which suggests cycles last from 1.5 to 8 years, we use $P_L=6$ and
# $P_H=32$ by default.

bk_cycles = sm.tsa.filters.bkfilter(dta[["infl", "unemp"]])

# * We lose K observations on both ends. It is suggested to use K=12 for
# quarterly data.

fig = plt.figure(figsize=(12, 10))
ax = fig.add_subplot(111)
bk_cycles.plot(ax=ax, style=["r--", "b-"])

# ### Christiano-Fitzgerald approximate band-pass filter: Inflation and
# Unemployment

# The Christiano-Fitzgerald filter is a generalization of BK and can thus
# also be seen as weighted moving average. However, the CF filter is
# asymmetric about $t$ as well as using the entire series. The
# implementation of their filter involves the
# calculations of the weights in
#
# $$y_{t}^{*}=B_{0}y_{t}+B_{1}y_{t+1}+\dots+B_{T-1-t}y_{T-1}+\tilde
# B_{T-t}y_{T}+B_{1}y_{t-1}+\dots+B_{t-2}y_{2}+\tilde B_{t-1}y_{1}$$
#
# for $t=3,4,...,T-2$, where
#
# $$B_{j} = \frac{\sin(jb)-\sin(ja)}{\pi j},j\geq1$$
#
# $$B_{0} = \frac{b-a}{\pi},a=\frac{2\pi}{P_{u}},b=\frac{2\pi}{P_{L}}$$
#
# $\tilde B_{T-t}$ and $\tilde B_{t-1}$ are linear functions of the
# $B_{j}$'s, and the values for $t=1,2,T-1,$ and $T$ are also calculated in
# much the same way. $P_{U}$ and $P_{L}$ are as described above with the
# same interpretation.

# The CF filter is appropriate for series that may follow a random walk.

print(sm.tsa.stattools.adfuller(dta["unemp"])[:3])

print(sm.tsa.stattools.adfuller(dta["infl"])[:3])

cf_cycles, cf_trend = sm.tsa.filters.cffilter(dta[["infl", "unemp"]])
print(cf_cycles.head(10))

fig = plt.figure(figsize=(14, 10))
ax = fig.add_subplot(111)
cf_cycles.plot(ax=ax, style=["r--", "b-"])

# Filtering assumes *a priori* that business cycles exist. Due to this
# assumption, many macroeconomic models seek to create models that match the
# shape of impulse response functions rather than replicating properties of
# filtered series. See VAR notebook.