Regression diagnostics

This example file shows how to use a few of the statsmodels regression diagnostic tests in a real-life context. You can learn about more tests and find out more information abou the tests here on the Regression Diagnostics page.

Note that most of the tests described here only return a tuple of numbers, without any annotation. A full description of outputs is always included in the docstring and in the online statsmodels documentation. For presentation purposes, we use the zip(name,test) construct to pretty-print short descriptions in the examples below.

Estimate a regression model

In [1]:
%matplotlib inline

from __future__ import print_function
from statsmodels.compat import lzip
import statsmodels
import numpy as np
import pandas as pd
import statsmodels.formula.api as smf
import statsmodels.stats.api as sms
import matplotlib.pyplot as plt

# Load data
url = 'http://vincentarelbundock.github.io/Rdatasets/csv/HistData/Guerry.csv'
dat = pd.read_csv(url)

# Fit regression model (using the natural log of one of the regressaors)
results = smf.ols('Lottery ~ Literacy + np.log(Pop1831)', data=dat).fit()

# Inspect the results
print(results.summary())
/build/statsmodels-hMUIZz/statsmodels-0.8.0/.pybuild/pythonX.Y_3.5/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
---------------------------------------------------------------------------
ConnectionRefusedError                    Traceback (most recent call last)
/usr/lib/python3.5/urllib/request.py in do_open(self, http_class, req, **http_conn_args)
   1253             try:
-> 1254                 h.request(req.get_method(), req.selector, req.data, headers)
   1255             except OSError as err: # timeout error

/usr/lib/python3.5/http/client.py in request(self, method, url, body, headers)
   1106         """Send a complete request to the server."""
-> 1107         self._send_request(method, url, body, headers)
   1108 

/usr/lib/python3.5/http/client.py in _send_request(self, method, url, body, headers)
   1151             body = _encode(body, 'body')
-> 1152         self.endheaders(body)
   1153 

/usr/lib/python3.5/http/client.py in endheaders(self, message_body)
   1102             raise CannotSendHeader()
-> 1103         self._send_output(message_body)
   1104 

/usr/lib/python3.5/http/client.py in _send_output(self, message_body)
    933 
--> 934         self.send(msg)
    935         if message_body is not None:

/usr/lib/python3.5/http/client.py in send(self, data)
    876             if self.auto_open:
--> 877                 self.connect()
    878             else:

/usr/lib/python3.5/http/client.py in connect(self)
    848         self.sock = self._create_connection(
--> 849             (self.host,self.port), self.timeout, self.source_address)
    850         self.sock.setsockopt(socket.IPPROTO_TCP, socket.TCP_NODELAY, 1)

/usr/lib/python3.5/socket.py in create_connection(address, timeout, source_address)
    711     if err is not None:
--> 712         raise err
    713     else:

/usr/lib/python3.5/socket.py in create_connection(address, timeout, source_address)
    702                 sock.bind(source_address)
--> 703             sock.connect(sa)
    704             return sock

ConnectionRefusedError: [Errno 111] Connection refused

During handling of the above exception, another exception occurred:

URLError                                  Traceback (most recent call last)
<ipython-input-1-9fe51e202f9f> in <module>()
     12 # Load data
     13 url = 'http://vincentarelbundock.github.io/Rdatasets/csv/HistData/Guerry.csv'
---> 14 dat = pd.read_csv(url)
     15 
     16 # Fit regression model (using the natural log of one of the regressaors)

/usr/lib/python3/dist-packages/pandas/io/parsers.py in parser_f(filepath_or_buffer, sep, delimiter, header, names, index_col, usecols, squeeze, prefix, mangle_dupe_cols, dtype, engine, converters, true_values, false_values, skipinitialspace, skiprows, nrows, na_values, keep_default_na, na_filter, verbose, skip_blank_lines, parse_dates, infer_datetime_format, keep_date_col, date_parser, dayfirst, iterator, chunksize, compression, thousands, decimal, lineterminator, quotechar, quoting, escapechar, comment, encoding, dialect, tupleize_cols, error_bad_lines, warn_bad_lines, skipfooter, skip_footer, doublequote, delim_whitespace, as_recarray, compact_ints, use_unsigned, low_memory, buffer_lines, memory_map, float_precision)
    653                     skip_blank_lines=skip_blank_lines)
    654 
--> 655         return _read(filepath_or_buffer, kwds)
    656 
    657     parser_f.__name__ = name

/usr/lib/python3/dist-packages/pandas/io/parsers.py in _read(filepath_or_buffer, kwds)
    390     compression = _infer_compression(filepath_or_buffer, compression)
    391     filepath_or_buffer, _, compression = get_filepath_or_buffer(
--> 392         filepath_or_buffer, encoding, compression)
    393     kwds['compression'] = compression
    394 

/usr/lib/python3/dist-packages/pandas/io/common.py in get_filepath_or_buffer(filepath_or_buffer, encoding, compression)
    184     if _is_url(filepath_or_buffer):
    185         url = str(filepath_or_buffer)
--> 186         req = _urlopen(url)
    187         content_encoding = req.headers.get('Content-Encoding', None)
    188         if content_encoding == 'gzip':

/usr/lib/python3.5/urllib/request.py in urlopen(url, data, timeout, cafile, capath, cadefault, context)
    161     else:
    162         opener = _opener
--> 163     return opener.open(url, data, timeout)
    164 
    165 def install_opener(opener):

/usr/lib/python3.5/urllib/request.py in open(self, fullurl, data, timeout)
    464             req = meth(req)
    465 
--> 466         response = self._open(req, data)
    467 
    468         # post-process response

/usr/lib/python3.5/urllib/request.py in _open(self, req, data)
    482         protocol = req.type
    483         result = self._call_chain(self.handle_open, protocol, protocol +
--> 484                                   '_open', req)
    485         if result:
    486             return result

/usr/lib/python3.5/urllib/request.py in _call_chain(self, chain, kind, meth_name, *args)
    442         for handler in handlers:
    443             func = getattr(handler, meth_name)
--> 444             result = func(*args)
    445             if result is not None:
    446                 return result

/usr/lib/python3.5/urllib/request.py in http_open(self, req)
   1280 
   1281     def http_open(self, req):
-> 1282         return self.do_open(http.client.HTTPConnection, req)
   1283 
   1284     http_request = AbstractHTTPHandler.do_request_

/usr/lib/python3.5/urllib/request.py in do_open(self, http_class, req, **http_conn_args)
   1254                 h.request(req.get_method(), req.selector, req.data, headers)
   1255             except OSError as err: # timeout error
-> 1256                 raise URLError(err)
   1257             r = h.getresponse()
   1258         except:

URLError: <urlopen error [Errno 111] Connection refused>

Normality of the residuals

Jarque-Bera test:

In [2]:
name = ['Jarque-Bera', 'Chi^2 two-tail prob.', 'Skew', 'Kurtosis']
test = sms.jarque_bera(results.resid)
lzip(name, test)
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-2-911bd7c54c5b> in <module>()
      1 name = ['Jarque-Bera', 'Chi^2 two-tail prob.', 'Skew', 'Kurtosis']
----> 2 test = sms.jarque_bera(results.resid)
      3 lzip(name, test)

NameError: name 'results' is not defined

Omni test:

In [3]:
name = ['Chi^2', 'Two-tail probability']
test = sms.omni_normtest(results.resid)
lzip(name, test)
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-3-5d66593e2f9b> in <module>()
      1 name = ['Chi^2', 'Two-tail probability']
----> 2 test = sms.omni_normtest(results.resid)
      3 lzip(name, test)

NameError: name 'results' is not defined

Influence tests

Once created, an object of class OLSInfluence holds attributes and methods that allow users to assess the influence of each observation. For example, we can compute and extract the first few rows of DFbetas by:

In [4]:
from statsmodels.stats.outliers_influence import OLSInfluence
test_class = OLSInfluence(results)
test_class.dfbetas[:5,:]
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-4-b1e3d0f42941> in <module>()
      1 from statsmodels.stats.outliers_influence import OLSInfluence
----> 2 test_class = OLSInfluence(results)
      3 test_class.dfbetas[:5,:]

NameError: name 'results' is not defined

Explore other options by typing dir(influence_test)

Useful information on leverage can also be plotted:

In [5]:
from statsmodels.graphics.regressionplots import plot_leverage_resid2
fig, ax = plt.subplots(figsize=(8,6))
fig = plot_leverage_resid2(results, ax = ax)
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-5-7590c2fd5fb2> in <module>()
      1 from statsmodels.graphics.regressionplots import plot_leverage_resid2
      2 fig, ax = plt.subplots(figsize=(8,6))
----> 3 fig = plot_leverage_resid2(results, ax = ax)

NameError: name 'results' is not defined

Other plotting options can be found on the Graphics page.

Multicollinearity

Condition number:

In [6]:
np.linalg.cond(results.model.exog)
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-6-c5af1dc95c76> in <module>()
----> 1 np.linalg.cond(results.model.exog)

NameError: name 'results' is not defined

Heteroskedasticity tests

Breush-Pagan test:

In [7]:
name = ['Lagrange multiplier statistic', 'p-value', 
        'f-value', 'f p-value']
test = sms.het_breushpagan(results.resid, results.model.exog)
lzip(name, test)
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-7-a4e5ba55306f> in <module>()
      1 name = ['Lagrange multiplier statistic', 'p-value', 
      2         'f-value', 'f p-value']
----> 3 test = sms.het_breushpagan(results.resid, results.model.exog)
      4 lzip(name, test)

NameError: name 'results' is not defined

Goldfeld-Quandt test

In [8]:
name = ['F statistic', 'p-value']
test = sms.het_goldfeldquandt(results.resid, results.model.exog)
lzip(name, test)
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-8-08d227eb273b> in <module>()
      1 name = ['F statistic', 'p-value']
----> 2 test = sms.het_goldfeldquandt(results.resid, results.model.exog)
      3 lzip(name, test)

NameError: name 'results' is not defined

Linearity

Harvey-Collier multiplier test for Null hypothesis that the linear specification is correct:

In [9]:
name = ['t value', 'p value']
test = sms.linear_harvey_collier(results)
lzip(name, test)
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-9-239f988c7dd2> in <module>()
      1 name = ['t value', 'p value']
----> 2 test = sms.linear_harvey_collier(results)
      3 lzip(name, test)

NameError: name 'results' is not defined