Regression diagnostics
======================


.. _regression_diagnostics_notebook:

`Link to Notebook GitHub <https://github.com/statsmodels/statsmodels/blob/master/examples/notebooks/regression_diagnostics.ipynb>`_

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   <p>This example file shows how to use a few of the <code>statsmodels</code> 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 <a href="http://statsmodels.sourceforge.net/stable/diagnostic.html">Regression Diagnostics page.</a> </p>
   <p>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 <code>statsmodels</code> documentation. For presentation purposes, we use the <code>zip(name,test)</code> construct to pretty-print short descriptions in the examples below.</p>
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   <h2 id="estimate-a-regression-model">Estimate a regression model</h2>
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   In&nbsp;[1]:
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   <div class="highlight"><pre><span class="kn">from</span> <span class="nn">__future__</span> <span class="kn">import</span> <span class="n">print_function</span>
   <span class="kn">from</span> <span class="nn">statsmodels.compat</span> <span class="kn">import</span> <span class="n">lzip</span>
   <span class="kn">import</span> <span class="nn">statsmodels</span>
   <span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
   <span class="kn">import</span> <span class="nn">pandas</span> <span class="kn">as</span> <span class="nn">pd</span>
   <span class="kn">import</span> <span class="nn">statsmodels.formula.api</span> <span class="kn">as</span> <span class="nn">smf</span>
   <span class="kn">import</span> <span class="nn">statsmodels.stats.api</span> <span class="kn">as</span> <span class="nn">sms</span>
   <span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
   
   <span class="c"># Load data</span>
   <span class="n">url</span> <span class="o">=</span> <span class="s">&#39;http://vincentarelbundock.github.io/Rdatasets/csv/HistData/Guerry.csv&#39;</span>
   <span class="n">dat</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="n">url</span><span class="p">)</span>
   
   <span class="c"># Fit regression model (using the natural log of one of the regressaors)</span>
   <span class="n">results</span> <span class="o">=</span> <span class="n">smf</span><span class="o">.</span><span class="n">ols</span><span class="p">(</span><span class="s">&#39;Lottery ~ Literacy + np.log(Pop1831)&#39;</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">dat</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
   
   <span class="c"># Inspect the results</span>
   <span class="k">print</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
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   <span class="ansired">URLError</span>                                  Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-56-5ab57e184000&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">     10</span> <span class="ansired"># Load data</span><span class="ansiblue"></span><span class="ansiblue"></span>
   <span class="ansigreen">     11</span> url <span class="ansiblue">=</span> <span class="ansiblue">&apos;http://vincentarelbundock.github.io/Rdatasets/csv/HistData/Guerry.csv&apos;</span><span class="ansiblue"></span>
   <span class="ansigreen">---&gt; 12</span><span class="ansired"> </span>dat <span class="ansiblue">=</span> pd<span class="ansiblue">.</span>read_csv<span class="ansiblue">(</span>url<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">     13</span> <span class="ansiblue"></span>
   <span class="ansigreen">     14</span> <span class="ansired"># Fit regression model (using the natural log of one of the regressaors)</span><span class="ansiblue"></span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/io/parsers.pyc</span> in <span class="ansicyan">parser_f</span><span class="ansiblue">(filepath_or_buffer, sep, dialect, compression, doublequote, escapechar, quotechar, quoting, skipinitialspace, lineterminator, header, index_col, names, prefix, skiprows, skipfooter, skip_footer, na_values, na_fvalues, true_values, false_values, delimiter, converters, dtype, usecols, engine, delim_whitespace, as_recarray, na_filter, compact_ints, use_unsigned, low_memory, buffer_lines, warn_bad_lines, error_bad_lines, keep_default_na, thousands, comment, decimal, parse_dates, keep_date_col, dayfirst, date_parser, memory_map, float_precision, nrows, iterator, chunksize, verbose, encoding, squeeze, mangle_dupe_cols, tupleize_cols, infer_datetime_format, skip_blank_lines)</span>
   <span class="ansigreen">    463</span>                     skip_blank_lines=skip_blank_lines)
   <span class="ansigreen">    464</span> <span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 465</span><span class="ansired">         </span><span class="ansigreen">return</span> _read<span class="ansiblue">(</span>filepath_or_buffer<span class="ansiblue">,</span> kwds<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    466</span> <span class="ansiblue"></span>
   <span class="ansigreen">    467</span>     parser_f<span class="ansiblue">.</span>__name__ <span class="ansiblue">=</span> name<span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/io/parsers.pyc</span> in <span class="ansicyan">_read</span><span class="ansiblue">(filepath_or_buffer, kwds)</span>
   <span class="ansigreen">    227</span> <span class="ansiblue"></span>
   <span class="ansigreen">    228</span>     filepath_or_buffer, _ = get_filepath_or_buffer(filepath_or_buffer,
   <span class="ansigreen">--&gt; 229</span><span class="ansired">                                                    encoding)
   </span><span class="ansigreen">    230</span> <span class="ansiblue"></span>
   <span class="ansigreen">    231</span>     <span class="ansigreen">if</span> kwds<span class="ansiblue">.</span>get<span class="ansiblue">(</span><span class="ansiblue">&apos;date_parser&apos;</span><span class="ansiblue">,</span> None<span class="ansiblue">)</span> <span class="ansigreen">is</span> <span class="ansigreen">not</span> None<span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/io/common.pyc</span> in <span class="ansicyan">get_filepath_or_buffer</span><span class="ansiblue">(filepath_or_buffer, encoding)</span>
   <span class="ansigreen">    116</span> <span class="ansiblue"></span>
   <span class="ansigreen">    117</span>     <span class="ansigreen">if</span> _is_url<span class="ansiblue">(</span>filepath_or_buffer<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 118</span><span class="ansired">         </span>req <span class="ansiblue">=</span> _urlopen<span class="ansiblue">(</span>str<span class="ansiblue">(</span>filepath_or_buffer<span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    119</span>         <span class="ansigreen">return</span> maybe_read_encoded_stream<span class="ansiblue">(</span>req<span class="ansiblue">,</span> encoding<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    120</span> <span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">urlopen</span><span class="ansiblue">(url, data, timeout, cafile, capath, cadefault, context)</span>
   <span class="ansigreen">    152</span>     <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    153</span>         opener <span class="ansiblue">=</span> _opener<span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 154</span><span class="ansired">     </span><span class="ansigreen">return</span> opener<span class="ansiblue">.</span>open<span class="ansiblue">(</span>url<span class="ansiblue">,</span> data<span class="ansiblue">,</span> timeout<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    155</span> <span class="ansiblue"></span>
   <span class="ansigreen">    156</span> <span class="ansigreen">def</span> install_opener<span class="ansiblue">(</span>opener<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">open</span><span class="ansiblue">(self, fullurl, data, timeout)</span>
   <span class="ansigreen">    429</span>             req <span class="ansiblue">=</span> meth<span class="ansiblue">(</span>req<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    430</span> <span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 431</span><span class="ansired">         </span>response <span class="ansiblue">=</span> self<span class="ansiblue">.</span>_open<span class="ansiblue">(</span>req<span class="ansiblue">,</span> data<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    432</span> <span class="ansiblue"></span>
   <span class="ansigreen">    433</span>         <span class="ansired"># post-process response</span><span class="ansiblue"></span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">_open</span><span class="ansiblue">(self, req, data)</span>
   <span class="ansigreen">    447</span>         protocol <span class="ansiblue">=</span> req<span class="ansiblue">.</span>get_type<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    448</span>         result = self._call_chain(self.handle_open, protocol, protocol +
   <span class="ansigreen">--&gt; 449</span><span class="ansired">                                   &apos;_open&apos;, req)
   </span><span class="ansigreen">    450</span>         <span class="ansigreen">if</span> result<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    451</span>             <span class="ansigreen">return</span> result<span class="ansiblue"></span>
   
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   <span class="ansigreen">    407</span>             func <span class="ansiblue">=</span> getattr<span class="ansiblue">(</span>handler<span class="ansiblue">,</span> meth_name<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    408</span> <span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 409</span><span class="ansired">             </span>result <span class="ansiblue">=</span> func<span class="ansiblue">(</span><span class="ansiblue">*</span>args<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    410</span>             <span class="ansigreen">if</span> result <span class="ansigreen">is</span> <span class="ansigreen">not</span> None<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    411</span>                 <span class="ansigreen">return</span> result<span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">http_open</span><span class="ansiblue">(self, req)</span>
   <span class="ansigreen">   1225</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1226</span>     <span class="ansigreen">def</span> http_open<span class="ansiblue">(</span>self<span class="ansiblue">,</span> req<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1227</span><span class="ansired">         </span><span class="ansigreen">return</span> self<span class="ansiblue">.</span>do_open<span class="ansiblue">(</span>httplib<span class="ansiblue">.</span>HTTPConnection<span class="ansiblue">,</span> req<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1228</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1229</span>     http_request <span class="ansiblue">=</span> AbstractHTTPHandler<span class="ansiblue">.</span>do_request_<span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">do_open</span><span class="ansiblue">(self, http_class, req, **http_conn_args)</span>
   <span class="ansigreen">   1195</span>         <span class="ansigreen">except</span> socket<span class="ansiblue">.</span>error<span class="ansiblue">,</span> err<span class="ansiblue">:</span> <span class="ansired"># XXX what error?</span><span class="ansiblue"></span>
   <span class="ansigreen">   1196</span>             h<span class="ansiblue">.</span>close<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1197</span><span class="ansired">             </span><span class="ansigreen">raise</span> URLError<span class="ansiblue">(</span>err<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1198</span>         <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">   1199</span>             <span class="ansigreen">try</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansired">URLError</span>: &lt;urlopen error [Errno -2] Name or service not known&gt;</pre>
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   <h2 id="normality-of-the-residuals">Normality of the residuals</h2>
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   <p>Jarque-Bera test:</p>
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   <div class="highlight"><pre><span class="n">name</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;Jarque-Bera&#39;</span><span class="p">,</span> <span class="s">&#39;Chi^2 two-tail prob.&#39;</span><span class="p">,</span> <span class="s">&#39;Skew&#39;</span><span class="p">,</span> <span class="s">&#39;Kurtosis&#39;</span><span class="p">]</span>
   <span class="n">test</span> <span class="o">=</span> <span class="n">sms</span><span class="o">.</span><span class="n">jarque_bera</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">resid</span><span class="p">)</span>
   <span class="n">lzip</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">test</span><span class="p">)</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-57-911bd7c54c5b&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">      1</span> name <span class="ansiblue">=</span> <span class="ansiblue">[</span><span class="ansiblue">&apos;Jarque-Bera&apos;</span><span class="ansiblue">,</span> <span class="ansiblue">&apos;Chi^2 two-tail prob.&apos;</span><span class="ansiblue">,</span> <span class="ansiblue">&apos;Skew&apos;</span><span class="ansiblue">,</span> <span class="ansiblue">&apos;Kurtosis&apos;</span><span class="ansiblue">]</span><span class="ansiblue"></span>
   <span class="ansigreen">----&gt; 2</span><span class="ansired"> </span>test <span class="ansiblue">=</span> sms<span class="ansiblue">.</span>jarque_bera<span class="ansiblue">(</span>results<span class="ansiblue">.</span>resid<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      3</span> lzip<span class="ansiblue">(</span>name<span class="ansiblue">,</span> test<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;results&apos; is not defined</pre>
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   <p>Omni test:</p>
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   <div class="highlight"><pre><span class="n">name</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;Chi^2&#39;</span><span class="p">,</span> <span class="s">&#39;Two-tail probability&#39;</span><span class="p">]</span>
   <span class="n">test</span> <span class="o">=</span> <span class="n">sms</span><span class="o">.</span><span class="n">omni_normtest</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">resid</span><span class="p">)</span>
   <span class="n">lzip</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">test</span><span class="p">)</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-58-5d66593e2f9b&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">      1</span> name <span class="ansiblue">=</span> <span class="ansiblue">[</span><span class="ansiblue">&apos;Chi^2&apos;</span><span class="ansiblue">,</span> <span class="ansiblue">&apos;Two-tail probability&apos;</span><span class="ansiblue">]</span><span class="ansiblue"></span>
   <span class="ansigreen">----&gt; 2</span><span class="ansired"> </span>test <span class="ansiblue">=</span> sms<span class="ansiblue">.</span>omni_normtest<span class="ansiblue">(</span>results<span class="ansiblue">.</span>resid<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      3</span> lzip<span class="ansiblue">(</span>name<span class="ansiblue">,</span> test<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;results&apos; is not defined</pre>
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   <h2 id="influence-tests">Influence tests</h2>
   <p>Once created, an object of class <code>OLSInfluence</code> 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:</p>
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   In&nbsp;[4]:
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   <div class="highlight"><pre><span class="kn">from</span> <span class="nn">statsmodels.stats.outliers_influence</span> <span class="kn">import</span> <span class="n">OLSInfluence</span>
   <span class="n">test_class</span> <span class="o">=</span> <span class="n">OLSInfluence</span><span class="p">(</span><span class="n">results</span><span class="p">)</span>
   <span class="n">test_class</span><span class="o">.</span><span class="n">dfbetas</span><span class="p">[:</span><span class="mi">5</span><span class="p">,:]</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-59-b1e3d0f42941&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">      1</span> <span class="ansigreen">from</span> statsmodels<span class="ansiblue">.</span>stats<span class="ansiblue">.</span>outliers_influence <span class="ansigreen">import</span> OLSInfluence<span class="ansiblue"></span>
   <span class="ansigreen">----&gt; 2</span><span class="ansired"> </span>test_class <span class="ansiblue">=</span> OLSInfluence<span class="ansiblue">(</span>results<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      3</span> test_class<span class="ansiblue">.</span>dfbetas<span class="ansiblue">[</span><span class="ansiblue">:</span><span class="ansicyan">5</span><span class="ansiblue">,</span><span class="ansiblue">:</span><span class="ansiblue">]</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;results&apos; is not defined</pre>
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   <div class="text_cell_render border-box-sizing rendered_html">
   <p>Explore other options by typing <code>dir(influence_test)</code></p>
   <p>Useful information on leverage can also be plotted:</p>
   </div>
   </div>
   </div>
   <div class="cell border-box-sizing code_cell rendered">
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   In&nbsp;[5]:
   </div>
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       <div class="input_area">
   <div class="highlight"><pre><span class="kn">from</span> <span class="nn">statsmodels.graphics.regressionplots</span> <span class="kn">import</span> <span class="n">plot_leverage_resid2</span>
   <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mi">6</span><span class="p">))</span>
   <span class="n">fig</span> <span class="o">=</span> <span class="n">plot_leverage_resid2</span><span class="p">(</span><span class="n">results</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">ax</span><span class="p">)</span>
   </pre></div>
   
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   <pre>
   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-60-7590c2fd5fb2&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">      1</span> <span class="ansigreen">from</span> statsmodels<span class="ansiblue">.</span>graphics<span class="ansiblue">.</span>regressionplots <span class="ansigreen">import</span> plot_leverage_resid2<span class="ansiblue"></span>
   <span class="ansigreen">      2</span> fig<span class="ansiblue">,</span> ax <span class="ansiblue">=</span> plt<span class="ansiblue">.</span>subplots<span class="ansiblue">(</span>figsize<span class="ansiblue">=</span><span class="ansiblue">(</span><span class="ansicyan">8</span><span class="ansiblue">,</span><span class="ansicyan">6</span><span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">----&gt; 3</span><span class="ansired"> </span>fig <span class="ansiblue">=</span> plot_leverage_resid2<span class="ansiblue">(</span>results<span class="ansiblue">,</span> ax <span class="ansiblue">=</span> ax<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;results&apos; is not defined</pre>
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   <p>Other plotting options can be found on the <a href="http://statsmodels.sourceforge.net/stable/graphics.html">Graphics page.</a></p>
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   <h2 id="multicollinearity">Multicollinearity</h2>
   <p>Condition number:</p>
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   In&nbsp;[6]:
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   <div class="highlight"><pre><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cond</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">exog</span><span class="p">)</span>
   </pre></div>
   
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   <pre>
   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-61-c5af1dc95c76&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>np<span class="ansiblue">.</span>linalg<span class="ansiblue">.</span>cond<span class="ansiblue">(</span>results<span class="ansiblue">.</span>model<span class="ansiblue">.</span>exog<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;results&apos; is not defined</pre>
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   <h2 id="heteroskedasticity-tests">Heteroskedasticity tests</h2>
   <p>Breush-Pagan test:</p>
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   In&nbsp;[7]:
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   <div class="highlight"><pre><span class="n">name</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;Lagrange multiplier statistic&#39;</span><span class="p">,</span> <span class="s">&#39;p-value&#39;</span><span class="p">,</span> 
           <span class="s">&#39;f-value&#39;</span><span class="p">,</span> <span class="s">&#39;f p-value&#39;</span><span class="p">]</span>
   <span class="n">test</span> <span class="o">=</span> <span class="n">sms</span><span class="o">.</span><span class="n">het_breushpagan</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">resid</span><span class="p">,</span> <span class="n">results</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">exog</span><span class="p">)</span>
   <span class="n">lzip</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">test</span><span class="p">)</span>
   </pre></div>
   
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-62-a4e5ba55306f&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">      1</span> name = [&apos;Lagrange multiplier statistic&apos;, &apos;p-value&apos;, 
   <span class="ansigreen">      2</span>         &apos;f-value&apos;, &apos;f p-value&apos;]
   <span class="ansigreen">----&gt; 3</span><span class="ansired"> </span>test <span class="ansiblue">=</span> sms<span class="ansiblue">.</span>het_breushpagan<span class="ansiblue">(</span>results<span class="ansiblue">.</span>resid<span class="ansiblue">,</span> results<span class="ansiblue">.</span>model<span class="ansiblue">.</span>exog<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      4</span> lzip<span class="ansiblue">(</span>name<span class="ansiblue">,</span> test<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;results&apos; is not defined</pre>
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   <p>Goldfeld-Quandt test</p>
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   In&nbsp;[8]:
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   <div class="highlight"><pre><span class="n">name</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;F statistic&#39;</span><span class="p">,</span> <span class="s">&#39;p-value&#39;</span><span class="p">]</span>
   <span class="n">test</span> <span class="o">=</span> <span class="n">sms</span><span class="o">.</span><span class="n">het_goldfeldquandt</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">resid</span><span class="p">,</span> <span class="n">results</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">exog</span><span class="p">)</span>
   <span class="n">lzip</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">test</span><span class="p">)</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-63-08d227eb273b&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">      1</span> name <span class="ansiblue">=</span> <span class="ansiblue">[</span><span class="ansiblue">&apos;F statistic&apos;</span><span class="ansiblue">,</span> <span class="ansiblue">&apos;p-value&apos;</span><span class="ansiblue">]</span><span class="ansiblue"></span>
   <span class="ansigreen">----&gt; 2</span><span class="ansired"> </span>test <span class="ansiblue">=</span> sms<span class="ansiblue">.</span>het_goldfeldquandt<span class="ansiblue">(</span>results<span class="ansiblue">.</span>resid<span class="ansiblue">,</span> results<span class="ansiblue">.</span>model<span class="ansiblue">.</span>exog<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      3</span> lzip<span class="ansiblue">(</span>name<span class="ansiblue">,</span> test<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;results&apos; is not defined</pre>
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   <h2 id="linearity">Linearity</h2>
   <p>Harvey-Collier multiplier test for Null hypothesis that the linear specification is correct:</p>
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   In&nbsp;[9]:
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   <div class="highlight"><pre><span class="n">name</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;t value&#39;</span><span class="p">,</span> <span class="s">&#39;p value&#39;</span><span class="p">]</span>
   <span class="n">test</span> <span class="o">=</span> <span class="n">sms</span><span class="o">.</span><span class="n">linear_harvey_collier</span><span class="p">(</span><span class="n">results</span><span class="p">)</span>
   <span class="n">lzip</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">test</span><span class="p">)</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-64-239f988c7dd2&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">      1</span> name <span class="ansiblue">=</span> <span class="ansiblue">[</span><span class="ansiblue">&apos;t value&apos;</span><span class="ansiblue">,</span> <span class="ansiblue">&apos;p value&apos;</span><span class="ansiblue">]</span><span class="ansiblue"></span>
   <span class="ansigreen">----&gt; 2</span><span class="ansired"> </span>test <span class="ansiblue">=</span> sms<span class="ansiblue">.</span>linear_harvey_collier<span class="ansiblue">(</span>results<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      3</span> lzip<span class="ansiblue">(</span>name<span class="ansiblue">,</span> test<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;results&apos; is not defined</pre>
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