%matplotlib inline

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);

"""
Explanation: Time Series Filters
End of explanation
"""


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);

"""
Explanation: 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
$$\min_{\{ \tau_{t}\} }\sum_{t}^{T}\zeta_{t}^{2}+\lambda\sum_{t=1}^{T}\left[\left(\tau_{t}-\tau_{t-1}\right)-\left(\tau_{t-1}-\tau_{t-2}\right)\right]^{2}$$
End of explanation
"""


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

"""
Explanation: 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.
End of explanation
"""


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

"""
Explanation: We lose K observations on both ends. It is suggested to use K=12 for quarterly data.
End of explanation
"""


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-']);

"""
Explanation: 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.
End of explanation
"""


