Ordinary Least SquaresΒΆ

Link to Notebook GitHub

In [1]:
from __future__ import print_function
import numpy as np
import statsmodels.api as sm
import matplotlib.pyplot as plt
from statsmodels.sandbox.regression.predstd import wls_prediction_std

np.random.seed(9876789)

OLS estimation

Artificial data:

In [2]:
nsample = 100
x = np.linspace(0, 10, 100)
X = np.column_stack((x, x**2))
beta = np.array([1, 0.1, 10])
e = np.random.normal(size=nsample)

Our model needs an intercept so we add a column of 1s:

In [3]:
X = sm.add_constant(X)
y = np.dot(X, beta) + e

Fit and summary:

In [4]:
model = sm.OLS(y, X)
results = model.fit()
print(results.summary())
                            OLS Regression Results
==============================================================================
Dep. Variable:                      y   R-squared:                       1.000
Model:                            OLS   Adj. R-squared:                  1.000
Method:                 Least Squares   F-statistic:                 4.020e+06
Date:                Wed, 27 Apr 2016   Prob (F-statistic):          2.83e-239
Time:                        01:53:58   Log-Likelihood:                -146.51
No. Observations:                 100   AIC:                             299.0
Df Residuals:                      97   BIC:                             306.8
Df Model:                           2
Covariance Type:            nonrobust
==============================================================================
                 coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
const          1.3423      0.313      4.292      0.000         0.722     1.963
x1            -0.0402      0.145     -0.278      0.781        -0.327     0.247
x2            10.0103      0.014    715.745      0.000         9.982    10.038
==============================================================================
Omnibus:                        2.042   Durbin-Watson:                   2.274
Prob(Omnibus):                  0.360   Jarque-Bera (JB):                1.875
Skew:                           0.234   Prob(JB):                        0.392
Kurtosis:                       2.519   Cond. No.                         144.
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Quantities of interest can be extracted directly from the fitted model. Type dir(results) for a full list. Here are some examples:

In [5]:
print('Parameters: ', results.params)
print('R2: ', results.rsquared)
Parameters:  [  1.3423  -0.0402  10.0103]
R2:  0.999987936503

OLS non-linear curve but linear in parameters

We simulate artificial data with a non-linear relationship between x and y:

In [6]:
nsample = 50
sig = 0.5
x = np.linspace(0, 20, nsample)
X = np.column_stack((x, np.sin(x), (x-5)**2, np.ones(nsample)))
beta = [0.5, 0.5, -0.02, 5.]

y_true = np.dot(X, beta)
y = y_true + sig * np.random.normal(size=nsample)

Fit and summary:

In [7]:
res = sm.OLS(y, X).fit()
print(res.summary())
                            OLS Regression Results
==============================================================================
Dep. Variable:                      y   R-squared:                       0.933
Model:                            OLS   Adj. R-squared:                  0.928
Method:                 Least Squares   F-statistic:                     211.8
Date:                Wed, 27 Apr 2016   Prob (F-statistic):           6.30e-27
Time:                        01:53:59   Log-Likelihood:                -34.438
No. Observations:                  50   AIC:                             76.88
Df Residuals:                      46   BIC:                             84.52
Df Model:                           3
Covariance Type:            nonrobust
==============================================================================
                 coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
x1             0.4687      0.026     17.751      0.000         0.416     0.522
x2             0.4836      0.104      4.659      0.000         0.275     0.693
x3            -0.0174      0.002     -7.507      0.000        -0.022    -0.013
const          5.2058      0.171     30.405      0.000         4.861     5.550
==============================================================================
Omnibus:                        0.655   Durbin-Watson:                   2.896
Prob(Omnibus):                  0.721   Jarque-Bera (JB):                0.360
Skew:                           0.207   Prob(JB):                        0.835
Kurtosis:                       3.026   Cond. No.                         221.
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Extract other quantities of interest:

In [8]:
print('Parameters: ', res.params)
print('Standard errors: ', res.bse)
print('Predicted values: ', res.predict())
Parameters:  [ 0.4687  0.4836 -0.0174  5.2058]
Standard errors:  [ 0.0264  0.1038  0.0023  0.1712]
Predicted values:  [  4.7707   5.2221   5.6362   5.9866   6.2564   6.4412   6.5493   6.6009
   6.6243   6.6518   6.7138   6.8341   7.0262   7.2905   7.6149   7.9763
   8.3446   8.6876   8.9764   9.19     9.3187   9.3659   9.3474   9.2889
   9.2217   9.1775   9.1834   9.2571   9.4044   9.6181   9.879   10.1591
  10.4266  10.6505  10.8063  10.8795  10.8683  10.7838  10.6483  10.4913
  10.3452  10.2393  10.1957  10.2249  10.3249  10.4808  10.6678  10.8549
  11.0101  11.1058]

Draw a plot to compare the true relationship to OLS predictions. Confidence intervals around the predictions are built using the wls_prediction_std command.

In [9]:
prstd, iv_l, iv_u = wls_prediction_std(res)

fig, ax = plt.subplots(figsize=(8,6))

ax.plot(x, y, 'o', label="data")
ax.plot(x, y_true, 'b-', label="True")
ax.plot(x, res.fittedvalues, 'r--.', label="OLS")
ax.plot(x, iv_u, 'r--')
ax.plot(x, iv_l, 'r--')
ax.legend(loc='best');
Error in callback <function post_execute at 0x7f89c987e578> (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 0x7f89b68ea550>

OLS with dummy variables

We generate some artificial data. There are 3 groups which will be modelled using dummy variables. Group 0 is the omitted/benchmark category.

In [10]:
nsample = 50
groups = np.zeros(nsample, int)
groups[20:40] = 1
groups[40:] = 2
#dummy = (groups[:,None] == np.unique(groups)).astype(float)

dummy = sm.categorical(groups, drop=True)
x = np.linspace(0, 20, nsample)
# drop reference category
X = np.column_stack((x, dummy[:,1:]))
X = sm.add_constant(X, prepend=False)

beta = [1., 3, -3, 10]
y_true = np.dot(X, beta)
e = np.random.normal(size=nsample)
y = y_true + e

Inspect the data:

In [11]:
print(X[:5,:])
print(y[:5])
print(groups)
print(dummy[:5,:])
[[ 0.      0.      0.      1.    ]
 [ 0.4082  0.      0.      1.    ]
 [ 0.8163  0.      0.      1.    ]
 [ 1.2245  0.      0.      1.    ]
 [ 1.6327  0.      0.      1.    ]]
[  9.2822  10.5048  11.8439  10.3851  12.3794]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 1 1 1 2 2 2 2 2 2 2 2 2 2]
[[ 1.  0.  0.]
 [ 1.  0.  0.]
 [ 1.  0.  0.]
 [ 1.  0.  0.]
 [ 1.  0.  0.]]

Fit and summary:

In [12]:
res2 = sm.OLS(y, X).fit()
print(res.summary())
                            OLS Regression Results
==============================================================================
Dep. Variable:                      y   R-squared:                       0.933
Model:                            OLS   Adj. R-squared:                  0.928
Method:                 Least Squares   F-statistic:                     211.8
Date:                Wed, 27 Apr 2016   Prob (F-statistic):           6.30e-27
Time:                        01:54:00   Log-Likelihood:                -34.438
No. Observations:                  50   AIC:                             76.88
Df Residuals:                      46   BIC:                             84.52
Df Model:                           3
Covariance Type:            nonrobust
==============================================================================
                 coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
x1             0.4687      0.026     17.751      0.000         0.416     0.522
x2             0.4836      0.104      4.659      0.000         0.275     0.693
x3            -0.0174      0.002     -7.507      0.000        -0.022    -0.013
const          5.2058      0.171     30.405      0.000         4.861     5.550
==============================================================================
Omnibus:                        0.655   Durbin-Watson:                   2.896
Prob(Omnibus):                  0.721   Jarque-Bera (JB):                0.360
Skew:                           0.207   Prob(JB):                        0.835
Kurtosis:                       3.026   Cond. No.                         221.
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Draw a plot to compare the true relationship to OLS predictions:

In [13]:
prstd, iv_l, iv_u = wls_prediction_std(res2)

fig, ax = plt.subplots(figsize=(8,6))

ax.plot(x, y, 'o', label="Data")
ax.plot(x, y_true, 'b-', label="True")
ax.plot(x, res2.fittedvalues, 'r--.', label="Predicted")
ax.plot(x, iv_u, 'r--')
ax.plot(x, iv_l, 'r--')
legend = ax.legend(loc="best")
Error in callback <function post_execute at 0x7f89c987e578> (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 0x7f89b64c3410>

Joint hypothesis test

F test

We want to test the hypothesis that both coefficients on the dummy variables are equal to zero, that is, $R \times \beta = 0$. An F test leads us to strongly reject the null hypothesis of identical constant in the 3 groups:

In [14]:
R = [[0, 1, 0, 0], [0, 0, 1, 0]]
print(np.array(R))
print(res2.f_test(R))
[[0 1 0 0]
 [0 0 1 0]]
<F test: F=array([[ 145.4927]]), p=1.28344196173e-20, df_denom=46, df_num=2>

You can also use formula-like syntax to test hypotheses

In [15]:
print(res2.f_test("x2 = x3 = 0"))
<F test: F=array([[ 145.4927]]), p=1.28344196173e-20, df_denom=46, df_num=2>

Small group effects

If we generate artificial data with smaller group effects, the T test can no longer reject the Null hypothesis:

In [16]:
beta = [1., 0.3, -0.0, 10]
y_true = np.dot(X, beta)
y = y_true + np.random.normal(size=nsample)

res3 = sm.OLS(y, X).fit()
In [17]:
print(res3.f_test(R))
<F test: F=array([[ 1.2249]]), p=0.303186441063, df_denom=46, df_num=2>

In [18]:
print(res3.f_test("x2 = x3 = 0"))
<F test: F=array([[ 1.2249]]), p=0.303186441063, df_denom=46, df_num=2>

Multicollinearity

The Longley dataset is well known to have high multicollinearity. That is, the exogenous predictors are highly correlated. This is problematic because it can affect the stability of our coefficient estimates as we make minor changes to model specification.

In [19]:
from statsmodels.datasets.longley import load_pandas
y = load_pandas().endog
X = load_pandas().exog
X = sm.add_constant(X)

Fit and summary:

In [20]:
ols_model = sm.OLS(y, X)
ols_results = ols_model.fit()
print(ols_results.summary())
                            OLS Regression Results
==============================================================================
Dep. Variable:                 TOTEMP   R-squared:                       0.995
Model:                            OLS   Adj. R-squared:                  0.992
Method:                 Least Squares   F-statistic:                     330.3
Date:                Wed, 27 Apr 2016   Prob (F-statistic):           4.98e-10
Time:                        01:54:02   Log-Likelihood:                -109.62
No. Observations:                  16   AIC:                             233.2
Df Residuals:                       9   BIC:                             238.6
Df Model:                           6
Covariance Type:            nonrobust
==============================================================================
                 coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
const      -3.482e+06    8.9e+05     -3.911      0.004      -5.5e+06 -1.47e+06
GNPDEFL       15.0619     84.915      0.177      0.863      -177.029   207.153
GNP           -0.0358      0.033     -1.070      0.313        -0.112     0.040
UNEMP         -2.0202      0.488     -4.136      0.003        -3.125    -0.915
ARMED         -1.0332      0.214     -4.822      0.001        -1.518    -0.549
POP           -0.0511      0.226     -0.226      0.826        -0.563     0.460
YEAR        1829.1515    455.478      4.016      0.003       798.788  2859.515
==============================================================================
Omnibus:                        0.749   Durbin-Watson:                   2.559
Prob(Omnibus):                  0.688   Jarque-Bera (JB):                0.684
Skew:                           0.420   Prob(JB):                        0.710
Kurtosis:                       2.434   Cond. No.                     4.86e+09
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 4.86e+09. This might indicate that there are
strong multicollinearity or other numerical problems.

/usr/lib/python2.7/dist-packages/scipy/stats/stats.py:1557: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=16
  "anyway, n=%i" % int(n))

Condition number

One way to assess multicollinearity is to compute the condition number. Values over 20 are worrisome (see Greene 4.9). The first step is to normalize the independent variables to have unit length:

In [21]:
norm_x = X.values
for i, name in enumerate(X):
    if name == "const":
        continue
    norm_x[:,i] = X[name]/np.linalg.norm(X[name])
norm_xtx = np.dot(norm_x.T,norm_x)

Then, we take the square root of the ratio of the biggest to the smallest eigen values.

In [22]:
eigs = np.linalg.eigvals(norm_xtx)
condition_number = np.sqrt(eigs.max() / eigs.min())
print(condition_number)
56240.8689362

Dropping an observation

Greene also points out that dropping a single observation can have a dramatic effect on the coefficient estimates:

In [23]:
ols_results2 = sm.OLS(y.ix[:14], X.ix[:14]).fit()
print("Percentage change %4.2f%%\n"*7 % tuple([i for i in (ols_results2.params - ols_results.params)/ols_results.params*100]))
Percentage change -13.35%
Percentage change -236.18%
Percentage change -23.69%
Percentage change -3.36%
Percentage change -7.26%
Percentage change -200.46%
Percentage change -13.34%


We can also look at formal statistics for this such as the DFBETAS -- a standardized measure of how much each coefficient changes when that observation is left out.

In [24]:
infl = ols_results.get_influence()

In general we may consider DBETAS in absolute value greater than $2/\sqrt{N}$ to be influential observations

In [25]:
2./len(X)**.5
Out[25]:
0.5
In [26]:
print(infl.summary_frame().filter(regex="dfb"))
    dfb_const  dfb_GNPDEFL   dfb_GNP  dfb_UNEMP  dfb_ARMED   dfb_POP  dfb_YEAR
0   -0.016406    -0.234566 -0.045095  -0.121513  -0.149026  0.211057  0.013388
1   -0.020608    -0.289091  0.124453   0.156964   0.287700 -0.161890  0.025958
2   -0.008382     0.007161 -0.016799   0.009575   0.002227  0.014871  0.008103
3    0.018093     0.907968 -0.500022  -0.495996   0.089996  0.711142 -0.040056
4    1.871260    -0.219351  1.611418   1.561520   1.169337 -1.081513 -1.864186
5   -0.321373    -0.077045 -0.198129  -0.192961  -0.430626  0.079916  0.323275
6    0.315945    -0.241983  0.438146   0.471797  -0.019546 -0.448515 -0.307517
7    0.015816    -0.002742  0.018591   0.005064  -0.031320 -0.015823 -0.015583
8   -0.004019    -0.045687  0.023708   0.018125   0.013683 -0.034770  0.005116
9   -1.018242    -0.282131 -0.412621  -0.663904  -0.715020 -0.229501  1.035723
10   0.030947    -0.024781  0.029480   0.035361   0.034508 -0.014194 -0.030805
11   0.005987    -0.079727  0.030276  -0.008883  -0.006854 -0.010693 -0.005323
12  -0.135883     0.092325 -0.253027  -0.211465   0.094720  0.331351  0.129120
13   0.032736    -0.024249  0.017510   0.033242   0.090655  0.007634 -0.033114
14   0.305868     0.148070  0.001428   0.169314   0.253431  0.342982 -0.318031
15  -0.538323     0.432004 -0.261262  -0.143444  -0.360890 -0.467296  0.552421