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-IFPJo1/statsmodels-0.8.0/.pybuild/cpython3_3.7_statsmodels/build/statsmodels/compat/pandas.py:56: FutureWarning: The pandas.core.datetools module is deprecated and will be removed in a future version. Please use the pandas.tseries module instead.
  from pandas.core import datetools
---------------------------------------------------------------------------
ConnectionRefusedError                    Traceback (most recent call last)
/usr/lib/python3.7/urllib/request.py in do_open(self, http_class, req, **http_conn_args)
   1316                 h.request(req.get_method(), req.selector, req.data, headers,
-> 1317                           encode_chunked=req.has_header('Transfer-encoding'))
   1318             except OSError as err: # timeout error

/usr/lib/python3.7/http/client.py in request(self, method, url, body, headers, encode_chunked)
   1228         """Send a complete request to the server."""
-> 1229         self._send_request(method, url, body, headers, encode_chunked)
   1230 

/usr/lib/python3.7/http/client.py in _send_request(self, method, url, body, headers, encode_chunked)
   1274             body = _encode(body, 'body')
-> 1275         self.endheaders(body, encode_chunked=encode_chunked)
   1276 

/usr/lib/python3.7/http/client.py in endheaders(self, message_body, encode_chunked)
   1223             raise CannotSendHeader()
-> 1224         self._send_output(message_body, encode_chunked=encode_chunked)
   1225 

/usr/lib/python3.7/http/client.py in _send_output(self, message_body, encode_chunked)
   1015         del self._buffer[:]
-> 1016         self.send(msg)
   1017 

/usr/lib/python3.7/http/client.py in send(self, data)
    955             if self.auto_open:
--> 956                 self.connect()
    957             else:

/usr/lib/python3.7/http/client.py in connect(self)
    927         self.sock = self._create_connection(
--> 928             (self.host,self.port), self.timeout, self.source_address)
    929         self.sock.setsockopt(socket.IPPROTO_TCP, socket.TCP_NODELAY, 1)

/usr/lib/python3.7/socket.py in create_connection(address, timeout, source_address)
    726     if err is not None:
--> 727         raise err
    728     else:

/usr/lib/python3.7/socket.py in create_connection(address, timeout, source_address)
    715                 sock.bind(source_address)
--> 716             sock.connect(sa)
    717             # Break explicitly a reference cycle

ConnectionRefusedError: [Errno 111] Connection refused

During handling of the above exception, another exception occurred:

URLError                                  Traceback (most recent call last)
<ipython-input-1-b87bc0f80fcc> 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, doublequote, delim_whitespace, low_memory, memory_map, float_precision)
    676                     skip_blank_lines=skip_blank_lines)
    677 
--> 678         return _read(filepath_or_buffer, kwds)
    679 
    680     parser_f.__name__ = name

/usr/lib/python3/dist-packages/pandas/io/parsers.py in _read(filepath_or_buffer, kwds)
    422     compression = _infer_compression(filepath_or_buffer, compression)
    423     filepath_or_buffer, _, compression, should_close = get_filepath_or_buffer(
--> 424         filepath_or_buffer, encoding, compression)
    425     kwds['compression'] = compression
    426 

/usr/lib/python3/dist-packages/pandas/io/common.py in get_filepath_or_buffer(filepath_or_buffer, encoding, compression, mode)
    193 
    194     if _is_url(filepath_or_buffer):
--> 195         req = _urlopen(filepath_or_buffer)
    196         content_encoding = req.headers.get('Content-Encoding', None)
    197         if content_encoding == 'gzip':

/usr/lib/python3.7/urllib/request.py in urlopen(url, data, timeout, cafile, capath, cadefault, context)
    220     else:
    221         opener = _opener
--> 222     return opener.open(url, data, timeout)
    223 
    224 def install_opener(opener):

/usr/lib/python3.7/urllib/request.py in open(self, fullurl, data, timeout)
    523             req = meth(req)
    524 
--> 525         response = self._open(req, data)
    526 
    527         # post-process response

/usr/lib/python3.7/urllib/request.py in _open(self, req, data)
    541         protocol = req.type
    542         result = self._call_chain(self.handle_open, protocol, protocol +
--> 543                                   '_open', req)
    544         if result:
    545             return result

/usr/lib/python3.7/urllib/request.py in _call_chain(self, chain, kind, meth_name, *args)
    501         for handler in handlers:
    502             func = getattr(handler, meth_name)
--> 503             result = func(*args)
    504             if result is not None:
    505                 return result

/usr/lib/python3.7/urllib/request.py in http_open(self, req)
   1343 
   1344     def http_open(self, req):
-> 1345         return self.do_open(http.client.HTTPConnection, req)
   1346 
   1347     http_request = AbstractHTTPHandler.do_request_

/usr/lib/python3.7/urllib/request.py in do_open(self, http_class, req, **http_conn_args)
   1317                           encode_chunked=req.has_header('Transfer-encoding'))
   1318             except OSError as err: # timeout error
-> 1319                 raise URLError(err)
   1320             r = h.getresponse()
   1321         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-ee73171ffab2> 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-3-91e9869152bc> 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-4-760546914e5a> 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-eb7313ad0e8d> 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-be81e152ec40> 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-7-86dbfc541fc0> 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-a9c0bcd95a9d> 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-a506880c3348> 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