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
/build/statsmodels-JytjB9/statsmodels-0.8.0/.pybuild/cpython3_3.6_statsmodels/build/statsmodels/compat/pandas.py:56: FutureWarning: The pandas.core.datetools module is deprecated and will be removed in a future version. Please use the pandas.tseries module instead.
  from pandas.core import datetools
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()
print(arma_mod20.params)
const                49.659366
ar.L1.SUNACTIVITY     1.390656
ar.L2.SUNACTIVITY    -0.688571
dtype: float64
/build/statsmodels-JytjB9/statsmodels-0.8.0/.pybuild/cpython3_3.6_statsmodels/build/statsmodels/tsa/kalmanf/kalmanfilter.py:646: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.
  if issubdtype(paramsdtype, float):
/build/statsmodels-JytjB9/statsmodels-0.8.0/.pybuild/cpython3_3.6_statsmodels/build/statsmodels/tsa/kalmanf/kalmanfilter.py:650: FutureWarning: Conversion of the second argument of issubdtype from `complex` to `np.complexfloating` is deprecated. In future, it will be treated as `np.complex128 == np.dtype(complex).type`.
  elif issubdtype(paramsdtype, complex):
In [9]:
arma_mod30 = sm.tsa.ARMA(dta, (3,0)).fit()
/build/statsmodels-JytjB9/statsmodels-0.8.0/.pybuild/cpython3_3.6_statsmodels/build/statsmodels/tsa/kalmanf/kalmanfilter.py:646: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.
  if issubdtype(paramsdtype, float):
/build/statsmodels-JytjB9/statsmodels-0.8.0/.pybuild/cpython3_3.6_statsmodels/build/statsmodels/tsa/kalmanf/kalmanfilter.py:650: FutureWarning: Conversion of the second argument of issubdtype from `complex` to `np.complexfloating` is deprecated. In future, it will be treated as `np.complex128 == np.dtype(complex).type`.
  elif issubdtype(paramsdtype, complex):
In [10]:
print(arma_mod20.aic, arma_mod20.bic, arma_mod20.hqic)
2622.636338063792 2637.569703171383 2628.6067259090382
/build/statsmodels-JytjB9/statsmodels-0.8.0/.pybuild/cpython3_3.6_statsmodels/build/statsmodels/tsa/kalmanf/kalmanfilter.py:646: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.
  if issubdtype(paramsdtype, float):
In [11]:
print(arma_mod30.params)
const                49.749796
ar.L1.SUNACTIVITY     1.300810
ar.L2.SUNACTIVITY    -0.508093
ar.L3.SUNACTIVITY    -0.129650
dtype: float64
In [12]:
print(arma_mod30.aic, arma_mod30.bic, arma_mod30.hqic)
2619.403628699364 2638.070335083853 2626.8666135059216
/build/statsmodels-JytjB9/statsmodels-0.8.0/.pybuild/cpython3_3.6_statsmodels/build/statsmodels/tsa/kalmanf/kalmanfilter.py:646: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.
  if issubdtype(paramsdtype, float):
  • Does our model obey the theory?
In [13]:
sm.stats.durbin_watson(arma_mod30.resid.values)
/build/statsmodels-JytjB9/statsmodels-0.8.0/.pybuild/cpython3_3.6_statsmodels/build/statsmodels/tsa/kalmanf/kalmanfilter.py:577: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.
  if issubdtype(paramsdtype, float):
Out[13]:
1.9564806905406757
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=49.845037117754686, pvalue=1.500678687416067e-11)
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'))
            AC          Q      Prob(>Q)
lag                                    
1.0   0.009179   0.026287  8.712008e-01
2.0   0.041793   0.573048  7.508690e-01
3.0  -0.001334   0.573607  9.024469e-01
4.0   0.136089   6.408955  1.706181e-01
5.0   0.092469   9.111881  1.046840e-01
6.0   0.091949  11.793308  6.674192e-02
7.0   0.068748  13.297273  6.518822e-02
8.0  -0.015020  13.369300  9.975911e-02
9.0   0.187592  24.641988  3.393810e-03
10.0  0.213718  39.322079  2.229398e-05
11.0  0.201082  52.361229  2.344860e-07
12.0  0.117182  56.804288  8.573905e-08
13.0 -0.014055  56.868423  1.893827e-07
14.0  0.015398  56.945664  3.997499e-07
15.0 -0.024967  57.149416  7.741175e-07
16.0  0.080916  59.296873  6.871888e-07
17.0  0.041138  59.853844  1.110899e-06
18.0 -0.052021  60.747533  1.548371e-06
19.0  0.062496  62.041796  1.831572e-06
20.0 -0.010301  62.077084  3.381114e-06
21.0  0.074453  63.926758  3.193467e-06
22.0  0.124955  69.154873  8.978021e-07
23.0  0.093162  72.071136  5.799571e-07
24.0 -0.082152  74.346787  4.712847e-07
25.0  0.015695  74.430143  8.288748e-07
26.0 -0.025037  74.643000  1.367237e-06
27.0 -0.125861  80.041239  3.722446e-07
28.0  0.053225  81.010073  4.716129e-07
29.0 -0.038693  81.523899  6.916415e-07
30.0 -0.016904  81.622318  1.151625e-06
31.0 -0.019296  81.751031  1.868708e-06
32.0  0.104990  85.575151  8.927700e-07
33.0  0.040086  86.134651  1.247474e-06
34.0  0.008829  86.161894  2.047769e-06
35.0  0.014588  86.236531  3.263719e-06
36.0 -0.119329  91.248977  1.084426e-06
37.0 -0.036665  91.723945  1.521884e-06
38.0 -0.046193  92.480592  1.938687e-06
39.0 -0.017768  92.592961  2.990607e-06
40.0 -0.006220  92.606784  4.696872e-06
  • 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    167.047353
1991-12-31    140.992848
1992-12-31     94.858876
1993-12-31     46.860614
1994-12-31     11.242300
1995-12-31     -4.721532
1996-12-31     -1.167078
1997-12-31     16.185598
1998-12-31     39.021839
1999-12-31     59.449843
2000-12-31     72.170093
2001-12-31     75.376689
2002-12-31     70.436311
2003-12-31     60.731396
2004-12-31     50.201584
2005-12-31     42.075817
2006-12-31     38.114099
2007-12-31     38.454487
2008-12-31     41.963690
2009-12-31     46.869181
2010-12-31     51.423160
2011-12-31     54.399610
2012-12-31     55.321566
Freq: A-DEC, dtype: float64
/build/statsmodels-JytjB9/statsmodels-0.8.0/.pybuild/cpython3_3.6_statsmodels/build/statsmodels/tsa/kalmanf/kalmanfilter.py:577: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.
  if issubdtype(paramsdtype, float):
In [21]:
fig, ax = plt.subplots(figsize=(12, 8))
ax = dta.ix['1950':].plot(ax=ax)
fig = arma_mod30.plot_predict('1990', '2012', dynamic=True, ax=ax, plot_insample=False)
/usr/lib/python3/dist-packages/ipykernel_launcher.py:2: DeprecationWarning: 
.ix is deprecated. Please use
.loc for label based indexing or
.iloc for positional indexing

See the documentation here:
http://pandas.pydata.org/pandas-docs/stable/indexing.html#ix-indexer-is-deprecated
  
/build/statsmodels-JytjB9/statsmodels-0.8.0/.pybuild/cpython3_3.6_statsmodels/build/statsmodels/tsa/kalmanf/kalmanfilter.py:577: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.
  if issubdtype(paramsdtype, float):
In [22]:
def mean_forecast_err(y, yhat):
    return y.sub(yhat).mean()
In [23]:
mean_forecast_err(dta.SUNACTIVITY, predict_sunspots)
Out[23]:
5.637114370345064

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()
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   32.687678  1.082211e-08
2.0  -0.172416   47.670747  4.450704e-11
3.0  -0.420945  137.159393  1.548466e-29
4.0  -0.046875  138.271302  6.617701e-29
5.0   0.103240  143.675909  2.958720e-29
6.0   0.214864  167.132999  1.823718e-33
7.0  -0.000889  167.133401  1.009206e-32
8.0  -0.045418  168.185753  3.094835e-32
9.0  -0.061445  170.115803  5.837214e-32
10.0  0.034623  170.729855  1.958737e-31
11.0  0.006351  170.750557  8.267052e-31
12.0 -0.012882  170.835910  3.220232e-30
13.0 -0.053959  172.336548  6.181195e-30
14.0 -0.016606  172.478965  2.160214e-29
15.0  0.051742  173.864488  4.089545e-29
16.0  0.078917  177.094281  3.217935e-29
17.0 -0.001834  177.096029  1.093167e-28
18.0 -0.101604  182.471938  3.103822e-29
19.0 -0.057342  184.187772  4.624065e-29
20.0  0.026975  184.568286  1.235670e-28
21.0  0.062359  186.605963  1.530258e-28
22.0 -0.009400  186.652365  4.548193e-28
23.0 -0.068037  189.088185  4.562009e-28
24.0 -0.035566  189.755202  9.901091e-28
25.0  0.095679  194.592623  3.354288e-28
26.0  0.065650  196.874878  3.487621e-28
27.0 -0.018404  197.054614  9.008745e-28
28.0 -0.079244  200.394009  5.773711e-28
29.0  0.008499  200.432502  1.541386e-27
30.0  0.053372  201.953776  2.133191e-27
31.0  0.074816  204.949395  1.550161e-27
32.0 -0.071187  207.667243  1.262288e-27
33.0 -0.088145  211.843156  5.480813e-28
34.0 -0.025283  212.187450  1.215227e-27
35.0  0.125690  220.714899  8.231607e-29
36.0  0.142724  231.734119  1.923081e-30
37.0  0.095768  236.706161  5.937778e-31
38.0 -0.084744  240.607804  2.890885e-31
39.0 -0.150126  252.878985  3.962997e-33
40.0 -0.083767  256.707742  1.996170e-33
/build/statsmodels-JytjB9/statsmodels-0.8.0/.pybuild/cpython3_3.6_statsmodels/build/statsmodels/tsa/kalmanf/kalmanfilter.py:646: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.
  if issubdtype(paramsdtype, float):
/build/statsmodels-JytjB9/statsmodels-0.8.0/.pybuild/cpython3_3.6_statsmodels/build/statsmodels/tsa/kalmanf/kalmanfilter.py:650: FutureWarning: Conversion of the second argument of issubdtype from `complex` to `np.complexfloating` is deprecated. In future, it will be treated as `np.complex128 == np.dtype(complex).type`.
  elif issubdtype(paramsdtype, complex):
/build/statsmodels-JytjB9/statsmodels-0.8.0/.pybuild/cpython3_3.6_statsmodels/build/statsmodels/tsa/kalmanf/kalmanfilter.py:577: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.
  if issubdtype(paramsdtype, float):
In [34]:
arma41 = sm.tsa.ARMA(arma_rvs, (4,1)).fit()
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'))
/build/statsmodels-JytjB9/statsmodels-0.8.0/.pybuild/cpython3_3.6_statsmodels/build/statsmodels/tsa/kalmanf/kalmanfilter.py:646: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.
  if issubdtype(paramsdtype, float):
/build/statsmodels-JytjB9/statsmodels-0.8.0/.pybuild/cpython3_3.6_statsmodels/build/statsmodels/tsa/kalmanf/kalmanfilter.py:650: FutureWarning: Conversion of the second argument of issubdtype from `complex` to `np.complexfloating` is deprecated. In future, it will be treated as `np.complex128 == np.dtype(complex).type`.
  elif issubdtype(paramsdtype, complex):
/build/statsmodels-JytjB9/statsmodels-0.8.0/.pybuild/cpython3_3.6_statsmodels/build/statsmodels/tsa/kalmanf/kalmanfilter.py:577: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.
  if issubdtype(paramsdtype, float):
            AC          Q  Prob(>Q)
lag                                
1.0  -0.007889   0.031303  0.859567
2.0   0.004132   0.039907  0.980244
3.0   0.018103   0.205418  0.976710
4.0  -0.006760   0.228543  0.993948
5.0   0.018120   0.395028  0.995465
6.0   0.050688   1.700453  0.945086
7.0   0.010252   1.753961  0.972196
8.0  -0.011206   1.818023  0.986091
9.0   0.020292   2.028522  0.991008
10.0  0.001029   2.029064  0.996113
11.0 -0.014035   2.130173  0.997984
12.0 -0.023858   2.422929  0.998427
13.0 -0.002108   2.425219  0.999339
14.0 -0.018783   2.607431  0.999590
15.0  0.011316   2.673700  0.999805
16.0  0.042159   3.595422  0.999443
17.0  0.007943   3.628208  0.999734
18.0 -0.074311   6.503855  0.993686
19.0 -0.023379   6.789067  0.995256
20.0  0.002398   6.792073  0.997313
21.0  0.000487   6.792198  0.998516
22.0  0.017953   6.961435  0.999024
23.0 -0.038576   7.744466  0.998744
24.0 -0.029816   8.213249  0.998859
25.0  0.077850  11.415821  0.990675
26.0  0.040408  12.280445  0.989479
27.0 -0.018612  12.464273  0.992262
28.0 -0.014764  12.580183  0.994586
29.0  0.017649  12.746187  0.996111
30.0 -0.005486  12.762260  0.997504
31.0  0.058256  14.578539  0.994614
32.0 -0.040840  15.473078  0.993887
33.0 -0.019493  15.677304  0.995393
34.0  0.037269  16.425461  0.995214
35.0  0.086212  20.437442  0.976296
36.0  0.041271  21.358840  0.974774
37.0  0.078704  24.716871  0.938949
38.0 -0.029729  25.197048  0.944895
39.0 -0.078397  28.543382  0.891179
40.0 -0.014466  28.657573  0.909268

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])
0.990432818833742