Autoregressive Moving Average (ARMA): Sunspots data¶
[1]:
%matplotlib inline
[2]:
import numpy as np
from scipy import stats
import pandas as pd
import matplotlib.pyplot as plt
import statsmodels.api as sm
[3]:
from statsmodels.graphics.api import qqplot
Sunspots Data¶
[4]:
print(sm.datasets.sunspots.NOTE)
::
Number of Observations - 309 (Annual 1700 - 2008)
Number of Variables - 1
Variable name definitions::
SUNACTIVITY - Number of sunspots for each year
The data file contains a 'YEAR' variable that is not returned by load.
[5]:
dta = sm.datasets.sunspots.load_pandas().data
[6]:
dta.index = pd.Index(sm.tsa.datetools.dates_from_range('1700', '2008'))
del dta["YEAR"]
[7]:
dta.plot(figsize=(12,8));

[8]:
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)

[9]:
arma_mod20 = sm.tsa.ARMA(dta, (2,0)).fit(disp=False)
print(arma_mod20.params)
const 49.659426
ar.L1.SUNACTIVITY 1.390656
ar.L2.SUNACTIVITY -0.688571
dtype: float64
/usr/lib/python3/dist-packages/statsmodels/tsa/base/tsa_model.py:159: ValueWarning: No frequency information was provided, so inferred frequency A-DEC will be used.
warnings.warn('No frequency information was'
[10]:
arma_mod30 = sm.tsa.ARMA(dta, (3,0)).fit(disp=False)
/usr/lib/python3/dist-packages/statsmodels/tsa/base/tsa_model.py:159: ValueWarning: No frequency information was provided, so inferred frequency A-DEC will be used.
warnings.warn('No frequency information was'
[11]:
print(arma_mod20.aic, arma_mod20.bic, arma_mod20.hqic)
2622.6363380637877 2637.5697031713785 2628.6067259090337
[12]:
print(arma_mod30.params)
const 49.749962
ar.L1.SUNACTIVITY 1.300810
ar.L2.SUNACTIVITY -0.508093
ar.L3.SUNACTIVITY -0.129650
dtype: float64
[13]:
print(arma_mod30.aic, arma_mod30.bic, arma_mod30.hqic)
2619.403628696482 2638.070335080971 2626.8666135030394
Does our model obey the theory?
[14]:
sm.stats.durbin_watson(arma_mod30.resid.values)
[14]:
1.9564808699288254
[15]:
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
ax = arma_mod30.resid.plot(ax=ax);

[16]:
resid = arma_mod30.resid
[17]:
stats.normaltest(resid)
[17]:
NormaltestResult(statistic=49.84501385314911, pvalue=1.5006961438664718e-11)
[18]:
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
fig = qqplot(resid, line='q', ax=ax, fit=True)

[19]:
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)

[20]:
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'))
AC Q Prob(>Q)
lag
1.0 0.009179 0.026286 8.712032e-01
2.0 0.041793 0.573039 7.508724e-01
3.0 -0.001335 0.573599 9.024488e-01
4.0 0.136089 6.408919 1.706205e-01
5.0 0.092468 9.111828 1.046860e-01
6.0 0.091948 11.793245 6.674343e-02
7.0 0.068748 13.297202 6.518981e-02
8.0 -0.015020 13.369230 9.976130e-02
9.0 0.187592 24.641907 3.393913e-03
10.0 0.213718 39.321990 2.229478e-05
11.0 0.201082 52.361130 2.344957e-07
12.0 0.117182 56.804179 8.574293e-08
13.0 -0.014055 56.868316 1.893910e-07
14.0 0.015398 56.945555 3.997674e-07
15.0 -0.024967 57.149309 7.741499e-07
16.0 0.080916 59.296762 6.872184e-07
17.0 0.041138 59.853731 1.110947e-06
18.0 -0.052021 60.747421 1.548436e-06
19.0 0.062496 62.041684 1.831648e-06
20.0 -0.010302 62.076971 3.381251e-06
21.0 0.074453 63.926644 3.193596e-06
22.0 0.124955 69.154761 8.978387e-07
23.0 0.093162 72.071023 5.799805e-07
24.0 -0.082152 74.346677 4.713033e-07
25.0 0.015695 74.430032 8.289070e-07
26.0 -0.025037 74.642891 1.367289e-06
27.0 -0.125861 80.041135 3.722581e-07
28.0 0.053225 81.009970 4.716296e-07
29.0 -0.038693 81.523795 6.916659e-07
30.0 -0.016904 81.622214 1.151665e-06
31.0 -0.019296 81.750927 1.868772e-06
32.0 0.104990 85.575052 8.927992e-07
33.0 0.040086 86.134554 1.247514e-06
34.0 0.008829 86.161797 2.047833e-06
35.0 0.014588 86.236434 3.263819e-06
36.0 -0.119329 91.248884 1.084458e-06
37.0 -0.036665 91.723852 1.521929e-06
38.0 -0.046193 92.480501 1.938742e-06
39.0 -0.017768 92.592869 2.990691e-06
40.0 -0.006220 92.606692 4.697001e-06
This indicates a lack of fit.
In-sample dynamic prediction. How good does our model do?
[21]:
predict_sunspots = arma_mod30.predict('1990', '2012', dynamic=True)
print(predict_sunspots)
1990-12-31 167.047431
1991-12-31 140.993034
1992-12-31 94.859162
1993-12-31 46.860956
1994-12-31 11.242636
1995-12-31 -4.721255
1996-12-31 -1.166890
1997-12-31 16.185700
1998-12-31 39.021885
1999-12-31 59.449876
2000-12-31 72.170156
2001-12-31 75.376808
2002-12-31 70.436491
2003-12-31 60.731624
2004-12-31 50.201834
2005-12-31 42.076060
2006-12-31 38.114314
2007-12-31 38.454663
2008-12-31 41.963831
2009-12-31 46.869301
2010-12-31 51.423276
2011-12-31 54.399737
2012-12-31 55.321713
Freq: A-DEC, dtype: float64
[22]:
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)

[23]:
def mean_forecast_err(y, yhat):
return y.sub(yhat).mean()
[24]:
mean_forecast_err(dta.SUNACTIVITY, predict_sunspots)
[24]:
5.636930696153281
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¶
[25]:
from statsmodels.tsa.arima_process import ArmaProcess
[26]:
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.
[27]:
arma_t = ArmaProcess(arparams, maparams)
[28]:
arma_t.isinvertible
[28]:
True
[29]:
arma_t.isstationary
[29]:
False
What does this mean?
[30]:
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
ax.plot(arma_t.generate_sample(nsample=50));

[31]:
arparams = np.array([1, .35, -.15, .55, .1])
maparams = np.array([1, .65])
arma_t = ArmaProcess(arparams, maparams)
arma_t.isstationary
[31]:
True
[32]:
arma_rvs = arma_t.generate_sample(nsample=500, burnin=250, scale=2.5)
[33]:
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.
[34]:
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'))
AC Q Prob(>Q)
lag
1.0 0.254921 32.687694 1.082202e-08
2.0 -0.172416 47.670772 4.450649e-11
3.0 -0.420945 137.159409 1.548453e-29
4.0 -0.046875 138.271320 6.617642e-29
5.0 0.103240 143.675931 2.958688e-29
6.0 0.214864 167.133017 1.823703e-33
7.0 -0.000889 167.133419 1.009197e-32
8.0 -0.045418 168.185772 3.094806e-32
9.0 -0.061445 170.115821 5.837164e-32
10.0 0.034623 170.729873 1.958720e-31
11.0 0.006351 170.750574 8.266983e-31
12.0 -0.012882 170.835927 3.220205e-30
13.0 -0.053959 172.336565 6.181144e-30
14.0 -0.016606 172.478983 2.160197e-29
15.0 0.051742 173.864506 4.089511e-29
16.0 0.078917 177.094299 3.217908e-29
17.0 -0.001834 177.096047 1.093158e-28
18.0 -0.101604 182.471956 3.103796e-29
19.0 -0.057342 184.187791 4.624025e-29
20.0 0.026975 184.568306 1.235659e-28
21.0 0.062359 186.605982 1.530245e-28
22.0 -0.009400 186.652384 4.548155e-28
23.0 -0.068037 189.088205 4.561969e-28
24.0 -0.035566 189.755221 9.901006e-28
25.0 0.095679 194.592642 3.354261e-28
26.0 0.065650 196.874897 3.487591e-28
27.0 -0.018404 197.054634 9.008666e-28
28.0 -0.079244 200.394029 5.773663e-28
29.0 0.008499 200.432521 1.541373e-27
30.0 0.053372 201.953796 2.133173e-27
31.0 0.074816 204.949414 1.550148e-27
32.0 -0.071187 207.667261 1.262278e-27
33.0 -0.088145 211.843176 5.480766e-28
34.0 -0.025283 212.187470 1.215217e-27
35.0 0.125690 220.714917 8.231541e-29
36.0 0.142724 231.734141 1.923063e-30
37.0 0.095768 236.706182 5.937724e-31
38.0 -0.084744 240.607825 2.890859e-31
39.0 -0.150126 252.879008 3.962957e-33
40.0 -0.083767 256.707765 1.996150e-33
[35]:
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'))
AC Q Prob(>Q)
lag
1.0 -0.007889 0.031302 0.859569
2.0 0.004132 0.039907 0.980244
3.0 0.018103 0.205418 0.976710
4.0 -0.006760 0.228541 0.993948
5.0 0.018120 0.395025 0.995466
6.0 0.050688 1.700445 0.945087
7.0 0.010252 1.753952 0.972197
8.0 -0.011206 1.818014 0.986092
9.0 0.020292 2.028515 0.991009
10.0 0.001029 2.029058 0.996113
11.0 -0.014035 2.130166 0.997984
12.0 -0.023858 2.422923 0.998427
13.0 -0.002108 2.425214 0.999339
14.0 -0.018783 2.607427 0.999590
15.0 0.011316 2.673696 0.999805
16.0 0.042159 3.595416 0.999443
17.0 0.007943 3.628201 0.999734
18.0 -0.074311 6.503850 0.993686
19.0 -0.023379 6.789062 0.995256
20.0 0.002398 6.792069 0.997313
21.0 0.000487 6.792193 0.998516
22.0 0.017952 6.961430 0.999024
23.0 -0.038576 7.744460 0.998744
24.0 -0.029816 8.213243 0.998859
25.0 0.077850 11.415817 0.990675
26.0 0.040408 12.280441 0.989479
27.0 -0.018612 12.464268 0.992262
28.0 -0.014764 12.580179 0.994586
29.0 0.017650 12.746183 0.996111
30.0 -0.005486 12.762256 0.997504
31.0 0.058256 14.578537 0.994614
32.0 -0.040840 15.473076 0.993887
33.0 -0.019493 15.677299 0.995393
34.0 0.037269 16.425456 0.995214
35.0 0.086212 20.437440 0.976296
36.0 0.041271 21.358837 0.974774
37.0 0.078704 24.716868 0.938949
38.0 -0.029729 25.197044 0.944895
39.0 -0.078397 28.543372 0.891179
40.0 -0.014466 28.657562 0.909269
Exercise: How good of in-sample prediction can you do for another series, say, CPI¶
[36]:
macrodta = sm.datasets.macrodata.load_pandas().data
macrodta.index = pd.Index(sm.tsa.datetools.dates_from_range('1959Q1', '2009Q3'))
cpi = macrodta["cpi"]
Hint:¶
[37]:
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.
[38]:
print(sm.tsa.adfuller(cpi)[1])
0.990432818833742