Autoregressive Moving Average (ARMA): Sunspots data

This notebook replicates the existing ARMA notebook using the statsmodels.tsa.statespace.SARIMAX class rather than the statsmodels.tsa.ARMA class.

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%matplotlib inline
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import numpy as np
from scipy import stats
import pandas as pd
import matplotlib.pyplot as plt

import statsmodels.api as sm
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from statsmodels.graphics.api import qqplot

Sunspots Data

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print(sm.datasets.sunspots.NOTE)
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dta = sm.datasets.sunspots.load_pandas().data
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dta.index = pd.Index(pd.date_range("1700", end="2009", freq="YE-DEC"))
del dta["YEAR"]
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dta.plot(figsize=(12,4));
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fig = plt.figure(figsize=(12,8))
ax1 = fig.add_subplot(211)
fig = sm.graphics.tsa.plot_acf(dta.values.squeeze(), lags=40, ax=ax1)
ax2 = fig.add_subplot(212)
fig = sm.graphics.tsa.plot_pacf(dta, lags=40, ax=ax2)
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arma_mod20 = sm.tsa.statespace.SARIMAX(dta, order=(2,0,0), trend='c').fit(disp=False)
print(arma_mod20.params)
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arma_mod30 = sm.tsa.statespace.SARIMAX(dta, order=(3,0,0), trend='c').fit(disp=False)
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print(arma_mod20.aic, arma_mod20.bic, arma_mod20.hqic)
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print(arma_mod30.params)
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print(arma_mod30.aic, arma_mod30.bic, arma_mod30.hqic)
  • Does our model obey the theory?

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sm.stats.durbin_watson(arma_mod30.resid)
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fig = plt.figure(figsize=(12,4))
ax = fig.add_subplot(111)
ax = plt.plot(arma_mod30.resid)
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resid = arma_mod30.resid
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stats.normaltest(resid)
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fig = plt.figure(figsize=(12,4))
ax = fig.add_subplot(111)
fig = qqplot(resid, line='q', ax=ax, fit=True)
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fig = plt.figure(figsize=(12,8))
ax1 = fig.add_subplot(211)
fig = sm.graphics.tsa.plot_acf(resid, lags=40, ax=ax1)
ax2 = fig.add_subplot(212)
fig = sm.graphics.tsa.plot_pacf(resid, lags=40, ax=ax2)
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r,q,p = sm.tsa.acf(resid, fft=True, qstat=True)
data = np.c_[r[1:], q, p]
index = pd.Index(range(1,q.shape[0]+1), name="lag")
table = pd.DataFrame(data, columns=["AC", "Q", "Prob(>Q)"], index=index)
print(table)
  • This indicates a lack of fit.

  • In-sample dynamic prediction. How good does our model do?

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predict_sunspots = arma_mod30.predict(start='1990', end='2012', dynamic=True)
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fig, ax = plt.subplots(figsize=(12, 8))
dta.loc['1950':].plot(ax=ax)
predict_sunspots.plot(ax=ax, style='r');
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def mean_forecast_err(y, yhat):
    return y.sub(yhat).mean()
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mean_forecast_err(dta.SUNACTIVITY, predict_sunspots)