Autoregressive Moving Average (ARMA): Sunspots data¶
In [1]:
%matplotlib inline
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
In [2]:
from statsmodels.graphics.api import qqplot
Sunpots Data¶
In [3]:
print(sm.datasets.sunspots.NOTE)
In [4]:
dta = sm.datasets.sunspots.load_pandas().data
In [5]:
dta.index = pd.Index(sm.tsa.datetools.dates_from_range('1700', '2008'))
del dta["YEAR"]
In [6]:
dta.plot(figsize=(12,8));
In [7]:
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)
In [8]:
arma_mod20 = sm.tsa.ARMA(dta, (2,0)).fit(disp=False)
print(arma_mod20.params)
In [9]:
arma_mod30 = sm.tsa.ARMA(dta, (3,0)).fit(disp=False)
In [10]:
print(arma_mod20.aic, arma_mod20.bic, arma_mod20.hqic)
In [11]:
print(arma_mod30.params)
In [12]:
print(arma_mod30.aic, arma_mod30.bic, arma_mod30.hqic)
- Does our model obey the theory?
In [13]:
sm.stats.durbin_watson(arma_mod30.resid.values)
Out[13]:
In [14]:
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
ax = arma_mod30.resid.plot(ax=ax);
In [15]:
resid = arma_mod30.resid
In [16]:
stats.normaltest(resid)
Out[16]:
In [17]:
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
fig = qqplot(resid, line='q', ax=ax, fit=True)
In [18]:
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)
In [19]:
r,q,p = sm.tsa.acf(resid.values.squeeze(), 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?
In [20]:
predict_sunspots = arma_mod30.predict('1990', '2012', dynamic=True)
print(predict_sunspots)
In [21]:
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)
In [22]:
def mean_forecast_err(y, yhat):
return y.sub(yhat).mean()
In [23]:
mean_forecast_err(dta.SUNACTIVITY, predict_sunspots)
Out[23]:
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¶
In [24]:
from statsmodels.tsa.arima_process import arma_generate_sample, ArmaProcess
In [25]:
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.
In [26]:
arma_t = ArmaProcess(arparams, maparams)
In [27]:
arma_t.isinvertible
Out[27]:
In [28]:
arma_t.isstationary
Out[28]:
- What does this mean?
In [29]:
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
ax.plot(arma_t.generate_sample(nsample=50));
In [30]:
arparams = np.array([1, .35, -.15, .55, .1])
maparams = np.array([1, .65])
arma_t = ArmaProcess(arparams, maparams)
arma_t.isstationary
Out[30]:
In [31]:
arma_rvs = arma_t.generate_sample(nsample=500, burnin=250, scale=2.5)
In [32]:
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.
In [33]:
arma11 = sm.tsa.ARMA(arma_rvs, (1,1)).fit(disp=False)
resid = arma11.resid
r,q,p = sm.tsa.acf(resid, 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'))
In [34]:
arma41 = sm.tsa.ARMA(arma_rvs, (4,1)).fit(disp=False)
resid = arma41.resid
r,q,p = sm.tsa.acf(resid, 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¶
In [35]:
macrodta = sm.datasets.macrodata.load_pandas().data
macrodta.index = pd.Index(sm.tsa.datetools.dates_from_range('1959Q1', '2009Q3'))
cpi = macrodta["cpi"]
Hint:¶
In [36]:
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, resoundly rejects the null of no unit-root.
In [37]:
print(sm.tsa.adfuller(cpi)[1])