Autoregressive Moving Average (ARMA): Sunspots data¶
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%matplotlib inline
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from __future__ import print_function
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
Sunpots 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(sm.tsa.datetools.dates_from_range('1700', '2008'))
del dta["YEAR"]
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dta.plot(figsize=(12,8));
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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.ARMA(dta, (2,0)).fit(disp=False)
print(arma_mod20.params)
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arma_mod30 = sm.tsa.ARMA(dta, (3,0)).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.values)
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fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
ax = arma_mod30.resid.plot(ax=ax);
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resid = arma_mod30.resid
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stats.normaltest(resid)
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fig = plt.figure(figsize=(12,8))
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.values.squeeze(), 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.values.squeeze(), fft=True, qstat=True)
data = np.c_[range(1,41), r[1:], q, p]
table = pd.DataFrame(data, columns=['lag', "AC", "Q", "Prob(>Q)"])
print(table.set_index('lag'))
- 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('1990', '2012', dynamic=True)
print(predict_sunspots)
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fig, ax = plt.subplots(figsize=(12, 8))
ax = dta.loc['1950':].plot(ax=ax)
fig = arma_mod30.plot_predict('1990', '2012', dynamic=True, ax=ax, plot_insample=False)
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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)
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Exercise: Can you obtain a better fit for the Sunspots model? (Hint: sm.tsa.AR has a method select_order)¶
Simulated ARMA(4,1): Model Identification is Difficult¶
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from statsmodels.tsa.arima_process import ArmaProcess
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np.random.seed(1234)
# include zero-th lag
arparams = np.array([1, .75, -.65, -.55, .9])
maparams = np.array([1, .65])
Let's make sure this model is estimable.
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arma_t = ArmaProcess(arparams, maparams)
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arma_t.isinvertible
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arma_t.isstationary
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- What does this mean?
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fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
ax.plot(arma_t.generate_sample(nsample=50));
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arparams = np.array([1, .35, -.15, .55, .1])
maparams = np.array([1, .65])
arma_t = ArmaProcess(arparams, maparams)
arma_t.isstationary
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arma_rvs = arma_t.generate_sample(nsample=500, burnin=250, scale=2.5)
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fig = plt.figure(figsize=(12,8))
ax1 = fig.add_subplot(211)
fig = sm.graphics.tsa.plot_acf(arma_rvs, lags=40, ax=ax1)
ax2 = fig.add_subplot(212)
fig = sm.graphics.tsa.plot_pacf(arma_rvs, lags=40, ax=ax2)
- For mixed ARMA processes the Autocorrelation function is a mixture of exponentials and damped sine waves after (q-p) lags.
- The partial autocorrelation function is a mixture of exponentials and dampened sine waves after (p-q) lags.
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arma11 = sm.tsa.ARMA(arma_rvs, (1,1)).fit(disp=False)
resid = arma11.resid
r,q,p = sm.tsa.acf(resid, fft=True, qstat=True)
data = np.c_[range(1,41), r[1:], q, p]
table = pd.DataFrame(data, columns=['lag', "AC", "Q", "Prob(>Q)"])
print(table.set_index('lag'))
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arma41 = sm.tsa.ARMA(arma_rvs, (4,1)).fit(disp=False)
resid = arma41.resid
r,q,p = sm.tsa.acf(resid, fft=True, qstat=True)
data = np.c_[range(1,41), r[1:], q, p]
table = pd.DataFrame(data, columns=['lag', "AC", "Q", "Prob(>Q)"])
print(table.set_index('lag'))
Exercise: How good of in-sample prediction can you do for another series, say, CPI¶
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macrodta = sm.datasets.macrodata.load_pandas().data
macrodta.index = pd.Index(sm.tsa.datetools.dates_from_range('1959Q1', '2009Q3'))
cpi = macrodta["cpi"]
Hint:¶
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fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
ax = cpi.plot(ax=ax);
ax.legend();
P-value of the unit-root test, resoundingly rejects the null of a unit-root.
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print(sm.tsa.adfuller(cpi)[1])