Autoregressive Moving Average (ARMA): Sunspots dataΒΆ

Link to Notebook GitHub

In [1]:
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));
Error in callback <function post_execute at 0x7ff5d7b48140> (for post_execute):

---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
/usr/lib/python2.7/dist-packages/matplotlib/pyplot.pyc in post_execute()
    145             def post_execute():
    146                 if matplotlib.is_interactive():
--> 147                     draw_all()
    148 
    149             # IPython >= 2

/usr/lib/python2.7/dist-packages/matplotlib/_pylab_helpers.pyc in draw_all(cls, force)
    148         for f_mgr in cls.get_all_fig_managers():
    149             if force or f_mgr.canvas.figure.stale:
--> 150                 f_mgr.canvas.draw_idle()
    151 
    152 atexit.register(Gcf.destroy_all)

/usr/lib/python2.7/dist-packages/matplotlib/backend_bases.pyc in draw_idle(self, *args, **kwargs)
   2024         if not self._is_idle_drawing:
   2025             with self._idle_draw_cntx():
-> 2026                 self.draw(*args, **kwargs)
   2027 
   2028     def draw_cursor(self, event):

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in draw(self)
    472 
    473         try:
--> 474             self.figure.draw(self.renderer)
    475         finally:
    476             RendererAgg.lock.release()

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/figure.pyc in draw(self, renderer)
   1157         dsu.sort(key=itemgetter(0))
   1158         for zorder, a, func, args in dsu:
-> 1159             func(*args)
   1160 
   1161         renderer.close_group('figure')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axes/_base.pyc in draw(self, renderer, inframe)
   2322 
   2323         for zorder, a in dsu:
-> 2324             a.draw(renderer)
   2325 
   2326         renderer.close_group('axes')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in draw(self, renderer, *args, **kwargs)
   1106         ticks_to_draw = self._update_ticks(renderer)
   1107         ticklabelBoxes, ticklabelBoxes2 = self._get_tick_bboxes(ticks_to_draw,
-> 1108                                                                 renderer)
   1109 
   1110         for tick in ticks_to_draw:

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in _get_tick_bboxes(self, ticks, renderer)
   1056         for tick in ticks:
   1057             if tick.label1On and tick.label1.get_visible():
-> 1058                 extent = tick.label1.get_window_extent(renderer)
   1059                 ticklabelBoxes.append(extent)
   1060             if tick.label2On and tick.label2.get_visible():

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in get_window_extent(self, renderer, dpi)
    959             raise RuntimeError('Cannot get window extent w/o renderer')
    960 
--> 961         bbox, info, descent = self._get_layout(self._renderer)
    962         x, y = self.get_unitless_position()
    963         x, y = self.get_transform().transform_point((x, y))

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in _get_layout(self, renderer)
    350         tmp, lp_h, lp_bl = renderer.get_text_width_height_descent('lp',
    351                                                          self._fontproperties,
--> 352                                                          ismath=False)
    353         offsety = (lp_h - lp_bl) * self._linespacing
    354 

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in get_text_width_height_descent(self, s, prop, ismath)
    227             fontsize = prop.get_size_in_points()
    228             w, h, d = texmanager.get_text_width_height_descent(s, fontsize,
--> 229                                                                renderer=self)
    230             return w, h, d
    231 

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in get_text_width_height_descent(self, tex, fontsize, renderer)
    673         else:
    674             # use dviread. It sometimes returns a wrong descent.
--> 675             dvifile = self.make_dvi(tex, fontsize)
    676             dvi = dviread.Dvi(dvifile, 72 * dpi_fraction)
    677             try:

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in make_dvi(self, tex, fontsize)
    420                      'string:\n%s\nHere is the full report generated by '
    421                      'LaTeX: \n\n' % repr(tex.encode('unicode_escape')) +
--> 422                      report))
    423             else:
    424                 mpl.verbose.report(report, 'debug')

RuntimeError: LaTeX was not able to process the following string:
'lp'
Here is the full report generated by LaTeX:

<matplotlib.figure.Figure at 0x7ff5c9599790>
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)
Error in callback <function post_execute at 0x7ff5d7b48140> (for post_execute):

---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
/usr/lib/python2.7/dist-packages/matplotlib/pyplot.pyc in post_execute()
    145             def post_execute():
    146                 if matplotlib.is_interactive():
--> 147                     draw_all()
    148 
    149             # IPython >= 2

/usr/lib/python2.7/dist-packages/matplotlib/_pylab_helpers.pyc in draw_all(cls, force)
    148         for f_mgr in cls.get_all_fig_managers():
    149             if force or f_mgr.canvas.figure.stale:
--> 150                 f_mgr.canvas.draw_idle()
    151 
    152 atexit.register(Gcf.destroy_all)

/usr/lib/python2.7/dist-packages/matplotlib/backend_bases.pyc in draw_idle(self, *args, **kwargs)
   2024         if not self._is_idle_drawing:
   2025             with self._idle_draw_cntx():
-> 2026                 self.draw(*args, **kwargs)
   2027 
   2028     def draw_cursor(self, event):

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in draw(self)
    472 
    473         try:
--> 474             self.figure.draw(self.renderer)
    475         finally:
    476             RendererAgg.lock.release()

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/figure.pyc in draw(self, renderer)
   1157         dsu.sort(key=itemgetter(0))
   1158         for zorder, a, func, args in dsu:
-> 1159             func(*args)
   1160 
   1161         renderer.close_group('figure')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axes/_base.pyc in draw(self, renderer, inframe)
   2322 
   2323         for zorder, a in dsu:
-> 2324             a.draw(renderer)
   2325 
   2326         renderer.close_group('axes')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in draw(self, renderer, *args, **kwargs)
   1106         ticks_to_draw = self._update_ticks(renderer)
   1107         ticklabelBoxes, ticklabelBoxes2 = self._get_tick_bboxes(ticks_to_draw,
-> 1108                                                                 renderer)
   1109 
   1110         for tick in ticks_to_draw:

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in _get_tick_bboxes(self, ticks, renderer)
   1056         for tick in ticks:
   1057             if tick.label1On and tick.label1.get_visible():
-> 1058                 extent = tick.label1.get_window_extent(renderer)
   1059                 ticklabelBoxes.append(extent)
   1060             if tick.label2On and tick.label2.get_visible():

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in get_window_extent(self, renderer, dpi)
    959             raise RuntimeError('Cannot get window extent w/o renderer')
    960 
--> 961         bbox, info, descent = self._get_layout(self._renderer)
    962         x, y = self.get_unitless_position()
    963         x, y = self.get_transform().transform_point((x, y))

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in _get_layout(self, renderer)
    350         tmp, lp_h, lp_bl = renderer.get_text_width_height_descent('lp',
    351                                                          self._fontproperties,
--> 352                                                          ismath=False)
    353         offsety = (lp_h - lp_bl) * self._linespacing
    354 

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in get_text_width_height_descent(self, s, prop, ismath)
    227             fontsize = prop.get_size_in_points()
    228             w, h, d = texmanager.get_text_width_height_descent(s, fontsize,
--> 229                                                                renderer=self)
    230             return w, h, d
    231 

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in get_text_width_height_descent(self, tex, fontsize, renderer)
    673         else:
    674             # use dviread. It sometimes returns a wrong descent.
--> 675             dvifile = self.make_dvi(tex, fontsize)
    676             dvi = dviread.Dvi(dvifile, 72 * dpi_fraction)
    677             try:

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in make_dvi(self, tex, fontsize)
    420                      'string:\n%s\nHere is the full report generated by '
    421                      'LaTeX: \n\n' % repr(tex.encode('unicode_escape')) +
--> 422                      report))
    423             else:
    424                 mpl.verbose.report(report, 'debug')

RuntimeError: LaTeX was not able to process the following string:
'lp'
Here is the full report generated by LaTeX:

<matplotlib.figure.Figure at 0x7ff5c91fc290>
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

In [9]:
arma_mod30 = sm.tsa.ARMA(dta, (3,0)).fit()
In [10]:
print(arma_mod20.aic, arma_mod20.bic, arma_mod20.hqic)
2622.63633806 2637.56970317 2628.60672591

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)
2619.4036287 2638.07033508 2626.86661351

  • Does our model obey the theory?
In [13]:
sm.stats.durbin_watson(arma_mod30.resid.values)
Out[13]:
1.9564806905406757
In [14]:
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
ax = arma_mod30.resid.plot(ax=ax);
Error in callback <function post_execute at 0x7ff5d7b48140> (for post_execute):

---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
/usr/lib/python2.7/dist-packages/matplotlib/pyplot.pyc in post_execute()
    145             def post_execute():
    146                 if matplotlib.is_interactive():
--> 147                     draw_all()
    148 
    149             # IPython >= 2

/usr/lib/python2.7/dist-packages/matplotlib/_pylab_helpers.pyc in draw_all(cls, force)
    148         for f_mgr in cls.get_all_fig_managers():
    149             if force or f_mgr.canvas.figure.stale:
--> 150                 f_mgr.canvas.draw_idle()
    151 
    152 atexit.register(Gcf.destroy_all)

/usr/lib/python2.7/dist-packages/matplotlib/backend_bases.pyc in draw_idle(self, *args, **kwargs)
   2024         if not self._is_idle_drawing:
   2025             with self._idle_draw_cntx():
-> 2026                 self.draw(*args, **kwargs)
   2027 
   2028     def draw_cursor(self, event):

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in draw(self)
    472 
    473         try:
--> 474             self.figure.draw(self.renderer)
    475         finally:
    476             RendererAgg.lock.release()

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/figure.pyc in draw(self, renderer)
   1157         dsu.sort(key=itemgetter(0))
   1158         for zorder, a, func, args in dsu:
-> 1159             func(*args)
   1160 
   1161         renderer.close_group('figure')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axes/_base.pyc in draw(self, renderer, inframe)
   2322 
   2323         for zorder, a in dsu:
-> 2324             a.draw(renderer)
   2325 
   2326         renderer.close_group('axes')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in draw(self, renderer, *args, **kwargs)
   1106         ticks_to_draw = self._update_ticks(renderer)
   1107         ticklabelBoxes, ticklabelBoxes2 = self._get_tick_bboxes(ticks_to_draw,
-> 1108                                                                 renderer)
   1109 
   1110         for tick in ticks_to_draw:

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in _get_tick_bboxes(self, ticks, renderer)
   1056         for tick in ticks:
   1057             if tick.label1On and tick.label1.get_visible():
-> 1058                 extent = tick.label1.get_window_extent(renderer)
   1059                 ticklabelBoxes.append(extent)
   1060             if tick.label2On and tick.label2.get_visible():

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in get_window_extent(self, renderer, dpi)
    959             raise RuntimeError('Cannot get window extent w/o renderer')
    960 
--> 961         bbox, info, descent = self._get_layout(self._renderer)
    962         x, y = self.get_unitless_position()
    963         x, y = self.get_transform().transform_point((x, y))

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in _get_layout(self, renderer)
    350         tmp, lp_h, lp_bl = renderer.get_text_width_height_descent('lp',
    351                                                          self._fontproperties,
--> 352                                                          ismath=False)
    353         offsety = (lp_h - lp_bl) * self._linespacing
    354 

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in get_text_width_height_descent(self, s, prop, ismath)
    227             fontsize = prop.get_size_in_points()
    228             w, h, d = texmanager.get_text_width_height_descent(s, fontsize,
--> 229                                                                renderer=self)
    230             return w, h, d
    231 

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in get_text_width_height_descent(self, tex, fontsize, renderer)
    673         else:
    674             # use dviread. It sometimes returns a wrong descent.
--> 675             dvifile = self.make_dvi(tex, fontsize)
    676             dvi = dviread.Dvi(dvifile, 72 * dpi_fraction)
    677             try:

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in make_dvi(self, tex, fontsize)
    420                      'string:\n%s\nHere is the full report generated by '
    421                      'LaTeX: \n\n' % repr(tex.encode('unicode_escape')) +
--> 422                      report))
    423             else:
    424                 mpl.verbose.report(report, 'debug')

RuntimeError: LaTeX was not able to process the following string:
'lp'
Here is the full report generated by LaTeX:

<matplotlib.figure.Figure at 0x7ff5c94e2fd0>
In [15]:
resid = arma_mod30.resid
In [16]:
stats.normaltest(resid)
Out[16]:
NormaltestResult(statistic=49.845037117754686, pvalue=1.5006786874160669e-11)
In [17]:
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
fig = qqplot(resid, line='q', ax=ax, fit=True)
Error in callback <function post_execute at 0x7ff5d7b48140> (for post_execute):

---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
/usr/lib/python2.7/dist-packages/matplotlib/pyplot.pyc in post_execute()
    145             def post_execute():
    146                 if matplotlib.is_interactive():
--> 147                     draw_all()
    148 
    149             # IPython >= 2

/usr/lib/python2.7/dist-packages/matplotlib/_pylab_helpers.pyc in draw_all(cls, force)
    148         for f_mgr in cls.get_all_fig_managers():
    149             if force or f_mgr.canvas.figure.stale:
--> 150                 f_mgr.canvas.draw_idle()
    151 
    152 atexit.register(Gcf.destroy_all)

/usr/lib/python2.7/dist-packages/matplotlib/backend_bases.pyc in draw_idle(self, *args, **kwargs)
   2024         if not self._is_idle_drawing:
   2025             with self._idle_draw_cntx():
-> 2026                 self.draw(*args, **kwargs)
   2027 
   2028     def draw_cursor(self, event):

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in draw(self)
    472 
    473         try:
--> 474             self.figure.draw(self.renderer)
    475         finally:
    476             RendererAgg.lock.release()

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/figure.pyc in draw(self, renderer)
   1157         dsu.sort(key=itemgetter(0))
   1158         for zorder, a, func, args in dsu:
-> 1159             func(*args)
   1160 
   1161         renderer.close_group('figure')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axes/_base.pyc in draw(self, renderer, inframe)
   2322 
   2323         for zorder, a in dsu:
-> 2324             a.draw(renderer)
   2325 
   2326         renderer.close_group('axes')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in draw(self, renderer, *args, **kwargs)
   1106         ticks_to_draw = self._update_ticks(renderer)
   1107         ticklabelBoxes, ticklabelBoxes2 = self._get_tick_bboxes(ticks_to_draw,
-> 1108                                                                 renderer)
   1109 
   1110         for tick in ticks_to_draw:

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in _get_tick_bboxes(self, ticks, renderer)
   1056         for tick in ticks:
   1057             if tick.label1On and tick.label1.get_visible():
-> 1058                 extent = tick.label1.get_window_extent(renderer)
   1059                 ticklabelBoxes.append(extent)
   1060             if tick.label2On and tick.label2.get_visible():

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in get_window_extent(self, renderer, dpi)
    959             raise RuntimeError('Cannot get window extent w/o renderer')
    960 
--> 961         bbox, info, descent = self._get_layout(self._renderer)
    962         x, y = self.get_unitless_position()
    963         x, y = self.get_transform().transform_point((x, y))

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in _get_layout(self, renderer)
    350         tmp, lp_h, lp_bl = renderer.get_text_width_height_descent('lp',
    351                                                          self._fontproperties,
--> 352                                                          ismath=False)
    353         offsety = (lp_h - lp_bl) * self._linespacing
    354 

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in get_text_width_height_descent(self, s, prop, ismath)
    227             fontsize = prop.get_size_in_points()
    228             w, h, d = texmanager.get_text_width_height_descent(s, fontsize,
--> 229                                                                renderer=self)
    230             return w, h, d
    231 

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in get_text_width_height_descent(self, tex, fontsize, renderer)
    673         else:
    674             # use dviread. It sometimes returns a wrong descent.
--> 675             dvifile = self.make_dvi(tex, fontsize)
    676             dvi = dviread.Dvi(dvifile, 72 * dpi_fraction)
    677             try:

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in make_dvi(self, tex, fontsize)
    420                      'string:\n%s\nHere is the full report generated by '
    421                      'LaTeX: \n\n' % repr(tex.encode('unicode_escape')) +
--> 422                      report))
    423             else:
    424                 mpl.verbose.report(report, 'debug')

RuntimeError: LaTeX was not able to process the following string:
'lp'
Here is the full report generated by LaTeX:

<matplotlib.figure.Figure at 0x7ff5c91fca50>
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)
Error in callback <function post_execute at 0x7ff5d7b48140> (for post_execute):

---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
/usr/lib/python2.7/dist-packages/matplotlib/pyplot.pyc in post_execute()
    145             def post_execute():
    146                 if matplotlib.is_interactive():
--> 147                     draw_all()
    148 
    149             # IPython >= 2

/usr/lib/python2.7/dist-packages/matplotlib/_pylab_helpers.pyc in draw_all(cls, force)
    148         for f_mgr in cls.get_all_fig_managers():
    149             if force or f_mgr.canvas.figure.stale:
--> 150                 f_mgr.canvas.draw_idle()
    151 
    152 atexit.register(Gcf.destroy_all)

/usr/lib/python2.7/dist-packages/matplotlib/backend_bases.pyc in draw_idle(self, *args, **kwargs)
   2024         if not self._is_idle_drawing:
   2025             with self._idle_draw_cntx():
-> 2026                 self.draw(*args, **kwargs)
   2027 
   2028     def draw_cursor(self, event):

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in draw(self)
    472 
    473         try:
--> 474             self.figure.draw(self.renderer)
    475         finally:
    476             RendererAgg.lock.release()

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/figure.pyc in draw(self, renderer)
   1157         dsu.sort(key=itemgetter(0))
   1158         for zorder, a, func, args in dsu:
-> 1159             func(*args)
   1160 
   1161         renderer.close_group('figure')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axes/_base.pyc in draw(self, renderer, inframe)
   2322 
   2323         for zorder, a in dsu:
-> 2324             a.draw(renderer)
   2325 
   2326         renderer.close_group('axes')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in draw(self, renderer, *args, **kwargs)
   1106         ticks_to_draw = self._update_ticks(renderer)
   1107         ticklabelBoxes, ticklabelBoxes2 = self._get_tick_bboxes(ticks_to_draw,
-> 1108                                                                 renderer)
   1109 
   1110         for tick in ticks_to_draw:

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in _get_tick_bboxes(self, ticks, renderer)
   1056         for tick in ticks:
   1057             if tick.label1On and tick.label1.get_visible():
-> 1058                 extent = tick.label1.get_window_extent(renderer)
   1059                 ticklabelBoxes.append(extent)
   1060             if tick.label2On and tick.label2.get_visible():

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in get_window_extent(self, renderer, dpi)
    959             raise RuntimeError('Cannot get window extent w/o renderer')
    960 
--> 961         bbox, info, descent = self._get_layout(self._renderer)
    962         x, y = self.get_unitless_position()
    963         x, y = self.get_transform().transform_point((x, y))

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in _get_layout(self, renderer)
    350         tmp, lp_h, lp_bl = renderer.get_text_width_height_descent('lp',
    351                                                          self._fontproperties,
--> 352                                                          ismath=False)
    353         offsety = (lp_h - lp_bl) * self._linespacing
    354 

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in get_text_width_height_descent(self, s, prop, ismath)
    227             fontsize = prop.get_size_in_points()
    228             w, h, d = texmanager.get_text_width_height_descent(s, fontsize,
--> 229                                                                renderer=self)
    230             return w, h, d
    231 

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in get_text_width_height_descent(self, tex, fontsize, renderer)
    673         else:
    674             # use dviread. It sometimes returns a wrong descent.
--> 675             dvifile = self.make_dvi(tex, fontsize)
    676             dvi = dviread.Dvi(dvifile, 72 * dpi_fraction)
    677             try:

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in make_dvi(self, tex, fontsize)
    420                      'string:\n%s\nHere is the full report generated by '
    421                      'LaTeX: \n\n' % repr(tex.encode('unicode_escape')) +
--> 422                      report))
    423             else:
    424                 mpl.verbose.report(report, 'debug')

RuntimeError: LaTeX was not able to process the following string:
'lp'
Here is the full report generated by LaTeX:

<matplotlib.figure.Figure at 0x7ff5c90e4850>
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    0.009179   0.026287  8.712008e-01
2    0.041793   0.573048  7.508690e-01
3   -0.001334   0.573607  9.024469e-01
4    0.136089   6.408955  1.706181e-01
5    0.092469   9.111881  1.046840e-01
6    0.091949  11.793308  6.674192e-02
7    0.068748  13.297273  6.518822e-02
8   -0.015020  13.369300  9.975911e-02
9    0.187592  24.641988  3.393810e-03
10   0.213718  39.322079  2.229398e-05
11   0.201082  52.361229  2.344860e-07
12   0.117182  56.804288  8.573905e-08
13  -0.014055  56.868423  1.893827e-07
14   0.015398  56.945664  3.997499e-07
15  -0.024967  57.149416  7.741175e-07
16   0.080916  59.296873  6.871888e-07
17   0.041138  59.853844  1.110899e-06
18  -0.052021  60.747533  1.548371e-06
19   0.062496  62.041796  1.831572e-06
20  -0.010301  62.077084  3.381114e-06
21   0.074453  63.926758  3.193467e-06
22   0.124955  69.154873  8.978021e-07
23   0.093162  72.071136  5.799571e-07
24  -0.082152  74.346787  4.712847e-07
25   0.015695  74.430143  8.288748e-07
26  -0.025037  74.643000  1.367237e-06
27  -0.125861  80.041239  3.722446e-07
28   0.053225  81.010073  4.716129e-07
29  -0.038693  81.523899  6.916415e-07
30  -0.016904  81.622318  1.151625e-06
31  -0.019296  81.751031  1.868708e-06
32   0.104990  85.575151  8.927700e-07
33   0.040086  86.134651  1.247474e-06
34   0.008829  86.161894  2.047769e-06
35   0.014588  86.236531  3.263719e-06
36  -0.119329  91.248977  1.084426e-06
37  -0.036665  91.723945  1.521884e-06
38  -0.046193  92.480592  1.938687e-06
39  -0.017768  92.592961  2.990607e-06
40  -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

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)
Error in callback <function post_execute at 0x7ff5d7b48140> (for post_execute):

---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
/usr/lib/python2.7/dist-packages/matplotlib/pyplot.pyc in post_execute()
    145             def post_execute():
    146                 if matplotlib.is_interactive():
--> 147                     draw_all()
    148 
    149             # IPython >= 2

/usr/lib/python2.7/dist-packages/matplotlib/_pylab_helpers.pyc in draw_all(cls, force)
    148         for f_mgr in cls.get_all_fig_managers():
    149             if force or f_mgr.canvas.figure.stale:
--> 150                 f_mgr.canvas.draw_idle()
    151 
    152 atexit.register(Gcf.destroy_all)

/usr/lib/python2.7/dist-packages/matplotlib/backend_bases.pyc in draw_idle(self, *args, **kwargs)
   2024         if not self._is_idle_drawing:
   2025             with self._idle_draw_cntx():
-> 2026                 self.draw(*args, **kwargs)
   2027 
   2028     def draw_cursor(self, event):

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in draw(self)
    472 
    473         try:
--> 474             self.figure.draw(self.renderer)
    475         finally:
    476             RendererAgg.lock.release()

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/figure.pyc in draw(self, renderer)
   1157         dsu.sort(key=itemgetter(0))
   1158         for zorder, a, func, args in dsu:
-> 1159             func(*args)
   1160 
   1161         renderer.close_group('figure')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axes/_base.pyc in draw(self, renderer, inframe)
   2322 
   2323         for zorder, a in dsu:
-> 2324             a.draw(renderer)
   2325 
   2326         renderer.close_group('axes')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in draw(self, renderer, *args, **kwargs)
   1106         ticks_to_draw = self._update_ticks(renderer)
   1107         ticklabelBoxes, ticklabelBoxes2 = self._get_tick_bboxes(ticks_to_draw,
-> 1108                                                                 renderer)
   1109 
   1110         for tick in ticks_to_draw:

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in _get_tick_bboxes(self, ticks, renderer)
   1056         for tick in ticks:
   1057             if tick.label1On and tick.label1.get_visible():
-> 1058                 extent = tick.label1.get_window_extent(renderer)
   1059                 ticklabelBoxes.append(extent)
   1060             if tick.label2On and tick.label2.get_visible():

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in get_window_extent(self, renderer, dpi)
    959             raise RuntimeError('Cannot get window extent w/o renderer')
    960 
--> 961         bbox, info, descent = self._get_layout(self._renderer)
    962         x, y = self.get_unitless_position()
    963         x, y = self.get_transform().transform_point((x, y))

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in _get_layout(self, renderer)
    350         tmp, lp_h, lp_bl = renderer.get_text_width_height_descent('lp',
    351                                                          self._fontproperties,
--> 352                                                          ismath=False)
    353         offsety = (lp_h - lp_bl) * self._linespacing
    354 

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in get_text_width_height_descent(self, s, prop, ismath)
    227             fontsize = prop.get_size_in_points()
    228             w, h, d = texmanager.get_text_width_height_descent(s, fontsize,
--> 229                                                                renderer=self)
    230             return w, h, d
    231 

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in get_text_width_height_descent(self, tex, fontsize, renderer)
    673         else:
    674             # use dviread. It sometimes returns a wrong descent.
--> 675             dvifile = self.make_dvi(tex, fontsize)
    676             dvi = dviread.Dvi(dvifile, 72 * dpi_fraction)
    677             try:

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in make_dvi(self, tex, fontsize)
    420                      'string:\n%s\nHere is the full report generated by '
    421                      'LaTeX: \n\n' % repr(tex.encode('unicode_escape')) +
--> 422                      report))
    423             else:
    424                 mpl.verbose.report(report, 'debug')

RuntimeError: LaTeX was not able to process the following string:
'lp'
Here is the full report generated by LaTeX:

<matplotlib.figure.Figure at 0x7ff5c912b290>
In [22]:
def mean_forecast_err(y, yhat):
    return y.sub(yhat).mean()
In [23]:
mean_forecast_err(dta.SUNACTIVITY, predict_sunspots)
Out[23]:
5.6371143703450644

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()
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-251-d3a1a0e5898b> in <module>()
----> 1 arma_t.isinvertible()

TypeError: 'bool' object is not callable
In [28]:
arma_t.isstationary()
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-252-55a9b2cc43b1> in <module>()
----> 1 arma_t.isstationary()

TypeError: 'bool' object is not callable
  • What does this mean?
In [29]:
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
ax.plot(arma_t.generate_sample(size=50));
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-253-d059f8971c1a> in <module>()
      1 fig = plt.figure(figsize=(12,8))
      2 ax = fig.add_subplot(111)
----> 3 ax.plot(arma_t.generate_sample(size=50));

TypeError: generate_sample() got an unexpected keyword argument 'size'
Error in callback <function post_execute at 0x7ff5d7b48140> (for post_execute):

---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
/usr/lib/python2.7/dist-packages/matplotlib/pyplot.pyc in post_execute()
    145             def post_execute():
    146                 if matplotlib.is_interactive():
--> 147                     draw_all()
    148 
    149             # IPython >= 2

/usr/lib/python2.7/dist-packages/matplotlib/_pylab_helpers.pyc in draw_all(cls, force)
    148         for f_mgr in cls.get_all_fig_managers():
    149             if force or f_mgr.canvas.figure.stale:
--> 150                 f_mgr.canvas.draw_idle()
    151 
    152 atexit.register(Gcf.destroy_all)

/usr/lib/python2.7/dist-packages/matplotlib/backend_bases.pyc in draw_idle(self, *args, **kwargs)
   2024         if not self._is_idle_drawing:
   2025             with self._idle_draw_cntx():
-> 2026                 self.draw(*args, **kwargs)
   2027 
   2028     def draw_cursor(self, event):

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in draw(self)
    472 
    473         try:
--> 474             self.figure.draw(self.renderer)
    475         finally:
    476             RendererAgg.lock.release()

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/figure.pyc in draw(self, renderer)
   1157         dsu.sort(key=itemgetter(0))
   1158         for zorder, a, func, args in dsu:
-> 1159             func(*args)
   1160 
   1161         renderer.close_group('figure')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axes/_base.pyc in draw(self, renderer, inframe)
   2322 
   2323         for zorder, a in dsu:
-> 2324             a.draw(renderer)
   2325 
   2326         renderer.close_group('axes')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in draw(self, renderer, *args, **kwargs)
   1106         ticks_to_draw = self._update_ticks(renderer)
   1107         ticklabelBoxes, ticklabelBoxes2 = self._get_tick_bboxes(ticks_to_draw,
-> 1108                                                                 renderer)
   1109 
   1110         for tick in ticks_to_draw:

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in _get_tick_bboxes(self, ticks, renderer)
   1056         for tick in ticks:
   1057             if tick.label1On and tick.label1.get_visible():
-> 1058                 extent = tick.label1.get_window_extent(renderer)
   1059                 ticklabelBoxes.append(extent)
   1060             if tick.label2On and tick.label2.get_visible():

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in get_window_extent(self, renderer, dpi)
    959             raise RuntimeError('Cannot get window extent w/o renderer')
    960 
--> 961         bbox, info, descent = self._get_layout(self._renderer)
    962         x, y = self.get_unitless_position()
    963         x, y = self.get_transform().transform_point((x, y))

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in _get_layout(self, renderer)
    350         tmp, lp_h, lp_bl = renderer.get_text_width_height_descent('lp',
    351                                                          self._fontproperties,
--> 352                                                          ismath=False)
    353         offsety = (lp_h - lp_bl) * self._linespacing
    354 

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in get_text_width_height_descent(self, s, prop, ismath)
    227             fontsize = prop.get_size_in_points()
    228             w, h, d = texmanager.get_text_width_height_descent(s, fontsize,
--> 229                                                                renderer=self)
    230             return w, h, d
    231 

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in get_text_width_height_descent(self, tex, fontsize, renderer)
    673         else:
    674             # use dviread. It sometimes returns a wrong descent.
--> 675             dvifile = self.make_dvi(tex, fontsize)
    676             dvi = dviread.Dvi(dvifile, 72 * dpi_fraction)
    677             try:

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in make_dvi(self, tex, fontsize)
    420                      'string:\n%s\nHere is the full report generated by '
    421                      'LaTeX: \n\n' % repr(tex.encode('unicode_escape')) +
--> 422                      report))
    423             else:
    424                 mpl.verbose.report(report, 'debug')

RuntimeError: LaTeX was not able to process the following string:
'lp'
Here is the full report generated by LaTeX:

<matplotlib.figure.Figure at 0x7ff5c912be90>
In [30]:
arparams = np.array([1, .35, -.15, .55, .1])
maparams = np.array([1, .65])
arma_t = ArmaProcess(arparams, maparams)
arma_t.isstationary()
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-254-317f1b2ac56b> in <module>()
      2 maparams = np.array([1, .65])
      3 arma_t = ArmaProcess(arparams, maparams)
----> 4 arma_t.isstationary()

TypeError: 'bool' object is not callable
In [31]:
arma_rvs = arma_t.generate_sample(size=500, burnin=250, scale=2.5)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-255-e0a3cc13cb6e> in <module>()
----> 1 arma_rvs = arma_t.generate_sample(size=500, burnin=250, scale=2.5)

TypeError: generate_sample() got an unexpected keyword argument 'size'
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)
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-256-8e761b44cfae> in <module>()
      1 fig = plt.figure(figsize=(12,8))
      2 ax1 = fig.add_subplot(211)
----> 3 fig = sm.graphics.tsa.plot_acf(arma_rvs, lags=40, ax=ax1)
      4 ax2 = fig.add_subplot(212)
      5 fig = sm.graphics.tsa.plot_pacf(arma_rvs, lags=40, ax=ax2)

NameError: name 'arma_rvs' is not defined
Error in callback <function post_execute at 0x7ff5d7b48140> (for post_execute):

---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
/usr/lib/python2.7/dist-packages/matplotlib/pyplot.pyc in post_execute()
    145             def post_execute():
    146                 if matplotlib.is_interactive():
--> 147                     draw_all()
    148 
    149             # IPython >= 2

/usr/lib/python2.7/dist-packages/matplotlib/_pylab_helpers.pyc in draw_all(cls, force)
    148         for f_mgr in cls.get_all_fig_managers():
    149             if force or f_mgr.canvas.figure.stale:
--> 150                 f_mgr.canvas.draw_idle()
    151 
    152 atexit.register(Gcf.destroy_all)

/usr/lib/python2.7/dist-packages/matplotlib/backend_bases.pyc in draw_idle(self, *args, **kwargs)
   2024         if not self._is_idle_drawing:
   2025             with self._idle_draw_cntx():
-> 2026                 self.draw(*args, **kwargs)
   2027 
   2028     def draw_cursor(self, event):

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in draw(self)
    472 
    473         try:
--> 474             self.figure.draw(self.renderer)
    475         finally:
    476             RendererAgg.lock.release()

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/figure.pyc in draw(self, renderer)
   1157         dsu.sort(key=itemgetter(0))
   1158         for zorder, a, func, args in dsu:
-> 1159             func(*args)
   1160 
   1161         renderer.close_group('figure')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axes/_base.pyc in draw(self, renderer, inframe)
   2322 
   2323         for zorder, a in dsu:
-> 2324             a.draw(renderer)
   2325 
   2326         renderer.close_group('axes')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in draw(self, renderer, *args, **kwargs)
   1106         ticks_to_draw = self._update_ticks(renderer)
   1107         ticklabelBoxes, ticklabelBoxes2 = self._get_tick_bboxes(ticks_to_draw,
-> 1108                                                                 renderer)
   1109 
   1110         for tick in ticks_to_draw:

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in _get_tick_bboxes(self, ticks, renderer)
   1056         for tick in ticks:
   1057             if tick.label1On and tick.label1.get_visible():
-> 1058                 extent = tick.label1.get_window_extent(renderer)
   1059                 ticklabelBoxes.append(extent)
   1060             if tick.label2On and tick.label2.get_visible():

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in get_window_extent(self, renderer, dpi)
    959             raise RuntimeError('Cannot get window extent w/o renderer')
    960 
--> 961         bbox, info, descent = self._get_layout(self._renderer)
    962         x, y = self.get_unitless_position()
    963         x, y = self.get_transform().transform_point((x, y))

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in _get_layout(self, renderer)
    350         tmp, lp_h, lp_bl = renderer.get_text_width_height_descent('lp',
    351                                                          self._fontproperties,
--> 352                                                          ismath=False)
    353         offsety = (lp_h - lp_bl) * self._linespacing
    354 

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in get_text_width_height_descent(self, s, prop, ismath)
    227             fontsize = prop.get_size_in_points()
    228             w, h, d = texmanager.get_text_width_height_descent(s, fontsize,
--> 229                                                                renderer=self)
    230             return w, h, d
    231 

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in get_text_width_height_descent(self, tex, fontsize, renderer)
    673         else:
    674             # use dviread. It sometimes returns a wrong descent.
--> 675             dvifile = self.make_dvi(tex, fontsize)
    676             dvi = dviread.Dvi(dvifile, 72 * dpi_fraction)
    677             try:

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in make_dvi(self, tex, fontsize)
    420                      'string:\n%s\nHere is the full report generated by '
    421                      'LaTeX: \n\n' % repr(tex.encode('unicode_escape')) +
--> 422                      report))
    423             else:
    424                 mpl.verbose.report(report, 'debug')

RuntimeError: LaTeX was not able to process the following string:
'lp'
Here is the full report generated by LaTeX:

<matplotlib.figure.Figure at 0x7ff5c94c8390>
  • 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'))
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-257-03653831c71c> in <module>()
----> 1 arma11 = sm.tsa.ARMA(arma_rvs, (1,1)).fit()
      2 resid = arma11.resid
      3 r,q,p = sm.tsa.acf(resid, qstat=True)
      4 data = np.c_[range(1,41), r[1:], q, p]
      5 table = pd.DataFrame(data, columns=['lag', "AC", "Q", "Prob(>Q)"])

NameError: name 'arma_rvs' is not defined
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'))
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-258-30d9c2f35894> in <module>()
----> 1 arma41 = sm.tsa.ARMA(arma_rvs, (4,1)).fit()
      2 resid = arma41.resid
      3 r,q,p = sm.tsa.acf(resid, qstat=True)
      4 data = np.c_[range(1,41), r[1:], q, p]
      5 table = pd.DataFrame(data, columns=['lag', "AC", "Q", "Prob(>Q)"])

NameError: name 'arma_rvs' is not defined

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();
Error in callback <function post_execute at 0x7ff5d7b48140> (for post_execute):

---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
/usr/lib/python2.7/dist-packages/matplotlib/pyplot.pyc in post_execute()
    145             def post_execute():
    146                 if matplotlib.is_interactive():
--> 147                     draw_all()
    148 
    149             # IPython >= 2

/usr/lib/python2.7/dist-packages/matplotlib/_pylab_helpers.pyc in draw_all(cls, force)
    148         for f_mgr in cls.get_all_fig_managers():
    149             if force or f_mgr.canvas.figure.stale:
--> 150                 f_mgr.canvas.draw_idle()
    151 
    152 atexit.register(Gcf.destroy_all)

/usr/lib/python2.7/dist-packages/matplotlib/backend_bases.pyc in draw_idle(self, *args, **kwargs)
   2024         if not self._is_idle_drawing:
   2025             with self._idle_draw_cntx():
-> 2026                 self.draw(*args, **kwargs)
   2027 
   2028     def draw_cursor(self, event):

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in draw(self)
    472 
    473         try:
--> 474             self.figure.draw(self.renderer)
    475         finally:
    476             RendererAgg.lock.release()

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/figure.pyc in draw(self, renderer)
   1157         dsu.sort(key=itemgetter(0))
   1158         for zorder, a, func, args in dsu:
-> 1159             func(*args)
   1160 
   1161         renderer.close_group('figure')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axes/_base.pyc in draw(self, renderer, inframe)
   2322 
   2323         for zorder, a in dsu:
-> 2324             a.draw(renderer)
   2325 
   2326         renderer.close_group('axes')

/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc in draw_wrapper(artist, renderer, *args, **kwargs)
     59     def draw_wrapper(artist, renderer, *args, **kwargs):
     60         before(artist, renderer)
---> 61         draw(artist, renderer, *args, **kwargs)
     62         after(artist, renderer)
     63 

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in draw(self, renderer, *args, **kwargs)
   1106         ticks_to_draw = self._update_ticks(renderer)
   1107         ticklabelBoxes, ticklabelBoxes2 = self._get_tick_bboxes(ticks_to_draw,
-> 1108                                                                 renderer)
   1109 
   1110         for tick in ticks_to_draw:

/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc in _get_tick_bboxes(self, ticks, renderer)
   1056         for tick in ticks:
   1057             if tick.label1On and tick.label1.get_visible():
-> 1058                 extent = tick.label1.get_window_extent(renderer)
   1059                 ticklabelBoxes.append(extent)
   1060             if tick.label2On and tick.label2.get_visible():

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in get_window_extent(self, renderer, dpi)
    959             raise RuntimeError('Cannot get window extent w/o renderer')
    960 
--> 961         bbox, info, descent = self._get_layout(self._renderer)
    962         x, y = self.get_unitless_position()
    963         x, y = self.get_transform().transform_point((x, y))

/usr/lib/python2.7/dist-packages/matplotlib/text.pyc in _get_layout(self, renderer)
    350         tmp, lp_h, lp_bl = renderer.get_text_width_height_descent('lp',
    351                                                          self._fontproperties,
--> 352                                                          ismath=False)
    353         offsety = (lp_h - lp_bl) * self._linespacing
    354 

/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc in get_text_width_height_descent(self, s, prop, ismath)
    227             fontsize = prop.get_size_in_points()
    228             w, h, d = texmanager.get_text_width_height_descent(s, fontsize,
--> 229                                                                renderer=self)
    230             return w, h, d
    231 

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in get_text_width_height_descent(self, tex, fontsize, renderer)
    673         else:
    674             # use dviread. It sometimes returns a wrong descent.
--> 675             dvifile = self.make_dvi(tex, fontsize)
    676             dvi = dviread.Dvi(dvifile, 72 * dpi_fraction)
    677             try:

/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc in make_dvi(self, tex, fontsize)
    420                      'string:\n%s\nHere is the full report generated by '
    421                      'LaTeX: \n\n' % repr(tex.encode('unicode_escape')) +
--> 422                      report))
    423             else:
    424                 mpl.verbose.report(report, 'debug')

RuntimeError: LaTeX was not able to process the following string:
'lp'
Here is the full report generated by LaTeX:

<matplotlib.figure.Figure at 0x7ff5c93059d0>

P-value of the unit-root test, resoundly rejects the null of no unit-root.

In [37]:
print(sm.tsa.adfuller(cpi)[1])
0.990432818834