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)
::

    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.

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)
const                49.659343
ar.L1.SUNACTIVITY     1.390656
ar.L2.SUNACTIVITY    -0.688571
dtype: float64
/home/travis/build/statsmodels/statsmodels/statsmodels/tsa/base/tsa_model.py:165: ValueWarning: No frequency information was provided, so inferred frequency A-DEC will be used.
  % freq, ValueWarning)
/home/travis/miniconda/envs/statsmodels-test/lib/python3.6/site-packages/scipy/signal/signaltools.py:1341: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use `arr[tuple(seq)]` instead of `arr[seq]`. In the future this will be interpreted as an array index, `arr[np.array(seq)]`, which will result either in an error or a different result.
  out_full[ind] += zi
/home/travis/miniconda/envs/statsmodels-test/lib/python3.6/site-packages/scipy/signal/signaltools.py:1344: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use `arr[tuple(seq)]` instead of `arr[seq]`. In the future this will be interpreted as an array index, `arr[np.array(seq)]`, which will result either in an error or a different result.
  out = out_full[ind]
/home/travis/miniconda/envs/statsmodels-test/lib/python3.6/site-packages/scipy/signal/signaltools.py:1350: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use `arr[tuple(seq)]` instead of `arr[seq]`. In the future this will be interpreted as an array index, `arr[np.array(seq)]`, which will result either in an error or a different result.
  zf = out_full[ind]
In [9]:
arma_mod30 = sm.tsa.ARMA(dta, (3,0)).fit(disp=False)
/home/travis/build/statsmodels/statsmodels/statsmodels/tsa/base/tsa_model.py:165: ValueWarning: No frequency information was provided, so inferred frequency A-DEC will be used.
  % freq, ValueWarning)
/home/travis/miniconda/envs/statsmodels-test/lib/python3.6/site-packages/scipy/signal/signaltools.py:1341: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use `arr[tuple(seq)]` instead of `arr[seq]`. In the future this will be interpreted as an array index, `arr[np.array(seq)]`, which will result either in an error or a different result.
  out_full[ind] += zi
/home/travis/miniconda/envs/statsmodels-test/lib/python3.6/site-packages/scipy/signal/signaltools.py:1344: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use `arr[tuple(seq)]` instead of `arr[seq]`. In the future this will be interpreted as an array index, `arr[np.array(seq)]`, which will result either in an error or a different result.
  out = out_full[ind]
/home/travis/miniconda/envs/statsmodels-test/lib/python3.6/site-packages/scipy/signal/signaltools.py:1350: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use `arr[tuple(seq)]` instead of `arr[seq]`. In the future this will be interpreted as an array index, `arr[np.array(seq)]`, which will result either in an error or a different result.
  zf = out_full[ind]
In [10]:
print(arma_mod20.aic, arma_mod20.bic, arma_mod20.hqic)
2622.6363380639814 2637.569703171572 2628.6067259092274
In [11]:
print(arma_mod30.params)
const                49.749900
ar.L1.SUNACTIVITY     1.300810
ar.L2.SUNACTIVITY    -0.508093
ar.L3.SUNACTIVITY    -0.129649
dtype: float64
In [12]:
print(arma_mod30.aic, arma_mod30.bic, arma_mod30.hqic)
2619.4036286966957 2638.0703350811846 2626.866613503253
  • Does our model obey the theory?
In [13]:
sm.stats.durbin_watson(arma_mod30.resid.values)
Out[13]:
1.9564809484851813
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]:
NormaltestResult(statistic=49.84502837307943, pvalue=1.500685248904302e-11)
In [17]:
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
fig = qqplot(resid, line='q', ax=ax, fit=True)
/home/travis/miniconda/envs/statsmodels-test/lib/python3.6/site-packages/scipy/stats/stats.py:1713: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use `arr[tuple(seq)]` instead of `arr[seq]`. In the future this will be interpreted as an array index, `arr[np.array(seq)]`, which will result either in an error or a different result.
  return np.add.reduce(sorted[indexer] * weights, axis=axis) / sumval
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'))
            AC          Q      Prob(>Q)
lag                                    
1.0   0.009179   0.026286  8.712033e-01
2.0   0.041793   0.573043  7.508710e-01
3.0  -0.001335   0.573602  9.024480e-01
4.0   0.136089   6.408918  1.706205e-01
5.0   0.092468   9.111822  1.046862e-01
6.0   0.091948  11.793234  6.674370e-02
7.0   0.068748  13.297190  6.519008e-02
8.0  -0.015020  13.369218  9.976168e-02
9.0   0.187592  24.641899  3.393923e-03
10.0  0.213718  39.321985  2.229482e-05
11.0  0.201082  52.361130  2.344956e-07
12.0  0.117182  56.804183  8.574277e-08
13.0 -0.014055  56.868320  1.893906e-07
14.0  0.015398  56.945559  3.997667e-07
15.0 -0.024967  57.149314  7.741483e-07
16.0  0.080916  59.296763  6.872181e-07
17.0  0.041138  59.853731  1.110947e-06
18.0 -0.052021  60.747422  1.548436e-06
19.0  0.062496  62.041685  1.831647e-06
20.0 -0.010301  62.076972  3.381250e-06
21.0  0.074453  63.926648  3.193592e-06
22.0  0.124955  69.154766  8.978368e-07
23.0  0.093162  72.071031  5.799790e-07
24.0 -0.082152  74.346683  4.713022e-07
25.0  0.015695  74.430039  8.289052e-07
26.0 -0.025037  74.642898  1.367285e-06
27.0 -0.125861  80.041145  3.722568e-07
28.0  0.053225  81.009978  4.716283e-07
29.0 -0.038693  81.523804  6.916637e-07
30.0 -0.016904  81.622223  1.151662e-06
31.0 -0.019296  81.750935  1.868767e-06
32.0  0.104990  85.575063  8.927962e-07
33.0  0.040086  86.134564  1.247509e-06
34.0  0.008829  86.161808  2.047826e-06
35.0  0.014588  86.236445  3.263808e-06
36.0 -0.119329  91.248896  1.084454e-06
37.0 -0.036665  91.723865  1.521923e-06
38.0 -0.046193  92.480514  1.938734e-06
39.0 -0.017768  92.592883  2.990678e-06
40.0 -0.006220  92.606706  4.696982e-06
/home/travis/build/statsmodels/statsmodels/statsmodels/tsa/stattools.py:529: FutureWarning: fft=True will become the default in a future version of statsmodels. To suppress this warning, explicitly set fft=False.
  warnings.warn(msg, FutureWarning)
  • 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)
1990-12-31    167.047399
1991-12-31    140.992964
1992-12-31     94.859065
1993-12-31     46.860852
1994-12-31     11.242545
1995-12-31     -4.721322
1996-12-31     -1.166932
1997-12-31     16.185671
1998-12-31     39.021857
1999-12-31     59.449835
2000-12-31     72.170096
2001-12-31     75.376731
2002-12-31     70.436404
2003-12-31     60.731536
2004-12-31     50.201753
2005-12-31     42.075992
2006-12-31     38.114258
2007-12-31     38.454616
2008-12-31     41.963786
2009-12-31     46.869253
2010-12-31     51.423221
2011-12-31     54.399673
2012-12-31     55.321643
Freq: A-DEC, dtype: float64
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]:
5.636994384847544

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]:
True
In [28]:
arma_t.isstationary
Out[28]:
False
  • 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]:
True
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'))
            AC           Q      Prob(>Q)
lag                                     
1.0   0.254921   32.687676  1.082212e-08
2.0  -0.172416   47.670745  4.450709e-11
3.0  -0.420945  137.159391  1.548467e-29
4.0  -0.046875  138.271300  6.617706e-29
5.0   0.103240  143.675907  2.958722e-29
6.0   0.214864  167.132997  1.823720e-33
7.0  -0.000889  167.133399  1.009206e-32
8.0  -0.045418  168.185751  3.094837e-32
9.0  -0.061445  170.115801  5.837219e-32
10.0  0.034623  170.729853  1.958738e-31
11.0  0.006351  170.750555  8.267060e-31
12.0 -0.012882  170.835908  3.220235e-30
13.0 -0.053959  172.336545  6.181201e-30
14.0 -0.016606  172.478963  2.160217e-29
15.0  0.051742  173.864486  4.089548e-29
16.0  0.078917  177.094279  3.217939e-29
17.0 -0.001834  177.096026  1.093169e-28
18.0 -0.101604  182.471936  3.103825e-29
19.0 -0.057342  184.187770  4.624070e-29
20.0  0.026975  184.568284  1.235672e-28
21.0  0.062359  186.605961  1.530260e-28
22.0 -0.009400  186.652363  4.548197e-28
23.0 -0.068037  189.088183  4.562015e-28
24.0 -0.035566  189.755200  9.901102e-28
25.0  0.095679  194.592622  3.354290e-28
26.0  0.065650  196.874876  3.487624e-28
27.0 -0.018404  197.054611  9.008754e-28
28.0 -0.079244  200.394008  5.773714e-28
29.0  0.008499  200.432501  1.541386e-27
30.0  0.053372  201.953774  2.133193e-27
31.0  0.074816  204.949395  1.550161e-27
32.0 -0.071187  207.667243  1.262287e-27
33.0 -0.088145  211.843155  5.480816e-28
34.0 -0.025283  212.187450  1.215228e-27
35.0  0.125690  220.714900  8.231603e-29
36.0  0.142724  231.734118  1.923082e-30
37.0  0.095768  236.706161  5.937778e-31
38.0 -0.084744  240.607805  2.890884e-31
39.0 -0.150126  252.878983  3.962999e-33
40.0 -0.083767  256.707741  1.996171e-33
/home/travis/build/statsmodels/statsmodels/statsmodels/tsa/stattools.py:529: FutureWarning: fft=True will become the default in a future version of statsmodels. To suppress this warning, explicitly set fft=False.
  warnings.warn(msg, FutureWarning)
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'))
            AC          Q  Prob(>Q)
lag                                
1.0  -0.007888   0.031300  0.859572
2.0   0.004132   0.039905  0.980245
3.0   0.018103   0.205413  0.976711
4.0  -0.006760   0.228536  0.993948
5.0   0.018120   0.395022  0.995466
6.0   0.050688   1.700444  0.945087
7.0   0.010252   1.753951  0.972197
8.0  -0.011206   1.818014  0.986092
9.0   0.020292   2.028515  0.991009
10.0  0.001029   2.029057  0.996113
11.0 -0.014035   2.130165  0.997984
12.0 -0.023858   2.422923  0.998427
13.0 -0.002108   2.425213  0.999339
14.0 -0.018783   2.607426  0.999590
15.0  0.011316   2.673696  0.999805
16.0  0.042159   3.595417  0.999443
17.0  0.007943   3.628203  0.999734
18.0 -0.074311   6.503854  0.993686
19.0 -0.023379   6.789066  0.995256
20.0  0.002398   6.792073  0.997313
21.0  0.000487   6.792197  0.998516
22.0  0.017952   6.961434  0.999024
23.0 -0.038576   7.744465  0.998744
24.0 -0.029816   8.213249  0.998859
25.0  0.077850  11.415825  0.990675
26.0  0.040408  12.280449  0.989479
27.0 -0.018612  12.464276  0.992262
28.0 -0.014764  12.580188  0.994586
29.0  0.017650  12.746192  0.996111
30.0 -0.005486  12.762265  0.997504
31.0  0.058256  14.578547  0.994614
32.0 -0.040840  15.473086  0.993887
33.0 -0.019493  15.677311  0.995393
34.0  0.037269  16.425467  0.995214
35.0  0.086212  20.437451  0.976296
36.0  0.041271  21.358850  0.974774
37.0  0.078704  24.716881  0.938948
38.0 -0.029729  25.197059  0.944895
39.0 -0.078397  28.543389  0.891178
40.0 -0.014466  28.657580  0.909268
/home/travis/build/statsmodels/statsmodels/statsmodels/tsa/stattools.py:529: FutureWarning: fft=True will become the default in a future version of statsmodels. To suppress this warning, explicitly set fft=False.
  warnings.warn(msg, FutureWarning)

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, resoundingly rejects the null of a unit-root.

In [37]:
print(sm.tsa.adfuller(cpi)[1])
0.9904328188337421