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<p><a class="reference internal" href="about.html#about-statsmodels"><span class="std std-ref">statsmodels</span></a> is a Python module that provides classes and functions for the estimation
of many different statistical models, as well as for conducting statistical tests, and statistical
data exploration. An extensive list of result statistics are available for each estimator.
The results are tested against existing statistical packages to ensure that they are correct. The
package is released under the open source Modified BSD (3-clause) license.
The online documentation is hosted at <a class="reference external" href="https://www.statsmodels.org/">statsmodels.org</a>.</p>
<h1 id="introduction">Introduction<a class="headerlink" href="#introduction" title="Link to this heading">¶</a></h1>
<p><code class="docutils literal notranslate"><span class="pre">statsmodels</span></code> supports specifying models using R-style formulas and <code class="docutils literal notranslate"><span class="pre">pandas</span></code> DataFrames.
Here is a simple example using ordinary least squares:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [1]: </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">In [2]: </span><span class="kn">import</span><span class="w"> </span><span class="nn">statsmodels.api</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">sm</span>
<span class="gp">In [3]: </span><span class="kn">import</span><span class="w"> </span><span class="nn">statsmodels.formula.api</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">smf</span>
<span class="go"># Load data</span>
<span class="gp">In [4]: </span><span class="n">dat</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">datasets</span><span class="o">.</span><span class="n">get_rdataset</span><span class="p">(</span><span class="s2">"Guerry"</span><span class="p">,</span> <span class="s2">"HistData"</span><span class="p">)</span><span class="o">.</span><span class="n">data</span>
<span class="go"># Fit regression model (using the natural log of one of the regressors)</span>
<span class="gp">In [5]: </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="s1">'Lottery ~ Literacy + np.log(Pop1831)'</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="go"># Inspect the results</span>
<span class="gp">In [6]: </span><span class="nb">print</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
<span class="go"> OLS Regression Results </span>
<span class="go">==============================================================================</span>
<span class="go">Dep. Variable: Lottery R-squared: 0.348</span>
<span class="go">Model: OLS Adj. R-squared: 0.333</span>
<span class="go">Method: Least Squares F-statistic: 22.20</span>
<span class="go">Date: Wed, 08 Apr 2026 Prob (F-statistic): 1.90e-08</span>
<span class="go">Time: 14:24:31 Log-Likelihood: -379.82</span>
<span class="go">No. Observations: 86 AIC: 765.6</span>
<span class="go">Df Residuals: 83 BIC: 773.0</span>
<span class="go">Df Model: 2 </span>
<span class="go">Covariance Type: nonrobust </span>
<span class="go">===================================================================================</span>
<span class="go"> coef std err t P>|t| [0.025 0.975]</span>
<span class="gt">-----------------------------------------------------------------------------------</span>
<span class="n">Intercept</span> <span class="mf">246.4341</span> <span class="mf">35.233</span> <span class="mf">6.995</span> <span class="mf">0.000</span> <span class="mf">176.358</span> <span class="mf">316.510</span>
<span class="n">Literacy</span> <span class="o">-</span><span class="mf">0.4889</span> <span class="mf">0.128</span> <span class="o">-</span><span class="mf">3.832</span> <span class="mf">0.000</span> <span class="o">-</span><span class="mf">0.743</span> <span class="o">-</span><span class="mf">0.235</span>
<span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">Pop1831</span><span class="p">)</span> <span class="o">-</span><span class="mf">31.3114</span> <span class="mf">5.977</span> <span class="o">-</span><span class="mf">5.239</span> <span class="mf">0.000</span> <span class="o">-</span><span class="mf">43.199</span> <span class="o">-</span><span class="mf">19.424</span>
<span class="o">==============================================================================</span>
<span class="ne">Omnibus</span>: 3.713 Durbin-Watson: 2.019
<span class="n">Prob</span><span class="p">(</span><span class="n">Omnibus</span><span class="p">):</span> <span class="mf">0.156</span> <span class="n">Jarque</span><span class="o">-</span><span class="n">Bera</span> <span class="p">(</span><span class="n">JB</span><span class="p">):</span> <span class="mf">3.394</span>
<span class="ne">Skew</span>: -0.487 Prob(JB): 0.183
<span class="ne">Kurtosis</span>: 3.003 Cond. No. 702.
<span class="o">==============================================================================</span>
<span class="ne">Notes</span>:
<span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="n">Standard</span> <span class="n">Errors</span> <span class="n">assume</span> <span class="n">that</span> <span class="n">the</span> <span class="n">covariance</span> <span class="n">matrix</span> <span class="n">of</span> <span class="n">the</span> <span class="n">errors</span> <span class="ow">is</span> <span class="n">correctly</span> <span class="n">specified</span><span class="o">.</span>
</code></pre></div>
</div>
<p>You can also use <code class="docutils literal notranslate"><span class="pre">numpy</span></code> arrays instead of formulas:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [7]: </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">In [8]: </span><span class="kn">import</span><span class="w"> </span><span class="nn">statsmodels.api</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">sm</span>
<span class="go"># Generate artificial data (2 regressors + constant)</span>
<span class="gp">In [9]: </span><span class="n">nobs</span> <span class="o">=</span> <span class="mi">100</span>
<span class="gp">In [10]: </span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">((</span><span class="n">nobs</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="gp">In [11]: </span><span class="n">X</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">add_constant</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="gp">In [12]: </span><span class="n">beta</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mf">.1</span><span class="p">,</span> <span class="mf">.5</span><span class="p">]</span>
<span class="gp">In [13]: </span><span class="n">e</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="n">nobs</span><span class="p">)</span>
<span class="gp">In [14]: </span><span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">beta</span><span class="p">)</span> <span class="o">+</span> <span class="n">e</span>
<span class="go"># Fit regression model</span>
<span class="gp">In [15]: </span><span class="n">results</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">OLS</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="go"># Inspect the results</span>
<span class="gp">In [16]: </span><span class="nb">print</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
<span class="go"> OLS Regression Results </span>
<span class="go">==============================================================================</span>
<span class="go">Dep. Variable: y R-squared: 0.247</span>
<span class="go">Model: OLS Adj. R-squared: 0.231</span>
<span class="go">Method: Least Squares F-statistic: 15.90</span>
<span class="go">Date: Wed, 08 Apr 2026 Prob (F-statistic): 1.07e-06</span>
<span class="go">Time: 14:24:31 Log-Likelihood: -18.185</span>
<span class="go">No. Observations: 100 AIC: 42.37</span>
<span class="go">Df Residuals: 97 BIC: 50.18</span>
<span class="go">Df Model: 2 </span>
<span class="go">Covariance Type: nonrobust </span>
<span class="go">==============================================================================</span>
<span class="go"> coef std err t P>|t| [0.025 0.975]</span>
<span class="gt">------------------------------------------------------------------------------</span>
<span class="n">const</span> <span class="mf">1.5135</span> <span class="mf">0.073</span> <span class="mf">20.685</span> <span class="mf">0.000</span> <span class="mf">1.368</span> <span class="mf">1.659</span>
<span class="n">x1</span> <span class="mf">0.1958</span> <span class="mf">0.102</span> <span class="mf">1.925</span> <span class="mf">0.057</span> <span class="o">-</span><span class="mf">0.006</span> <span class="mf">0.398</span>
<span class="n">x2</span> <span class="mf">0.4922</span> <span class="mf">0.104</span> <span class="mf">4.740</span> <span class="mf">0.000</span> <span class="mf">0.286</span> <span class="mf">0.698</span>
<span class="o">==============================================================================</span>
<span class="ne">Omnibus</span>: 23.831 Durbin-Watson: 1.951
<span class="n">Prob</span><span class="p">(</span><span class="n">Omnibus</span><span class="p">):</span> <span class="mf">0.000</span> <span class="n">Jarque</span><span class="o">-</span><span class="n">Bera</span> <span class="p">(</span><span class="n">JB</span><span class="p">):</span> <span class="mf">6.295</span>
<span class="ne">Skew</span>: -0.262 Prob(JB): 0.0430
<span class="ne">Kurtosis</span>: 1.888 Cond. No. 4.95
<span class="o">==============================================================================</span>
<span class="ne">Notes</span>:
<span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="n">Standard</span> <span class="n">Errors</span> <span class="n">assume</span> <span class="n">that</span> <span class="n">the</span> <span class="n">covariance</span> <span class="n">matrix</span> <span class="n">of</span> <span class="n">the</span> <span class="n">errors</span> <span class="ow">is</span> <span class="n">correctly</span> <span class="n">specified</span><span class="o">.</span>
</code></pre></div>
</div>
<p>Have a look at <cite>dir(results)</cite> to see available results. Attributes are described in
<cite>results.__doc__</cite> and results methods have their own docstrings.</p>
<h1 id="citation">Citation<a class="headerlink" href="#citation" title="Link to this heading">¶</a></h1>
<p>Please use following citation to cite statsmodels in scientific publications:</p>
<p>Seabold, Skipper, and Josef Perktold. “<a class="reference external" href="https://proceedings.scipy.org/articles/proceedings-2010.pdf">statsmodels: Econometric and statistical modeling with
python.</a>” <em>Proceedings
of the 9th Python in Science Conference.</em> 2010.</p>
<p>Bibtex entry:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><code><span class="nd">@inproceedings</span><span class="p">{</span><span class="n">seabold2010statsmodels</span><span class="p">,</span>
<span class="n">title</span><span class="o">=</span><span class="p">{</span><span class="n">statsmodels</span><span class="p">:</span> <span class="n">Econometric</span> <span class="ow">and</span> <span class="n">statistical</span> <span class="n">modeling</span> <span class="k">with</span> <span class="n">python</span><span class="p">},</span>
<span class="n">author</span><span class="o">=</span><span class="p">{</span><span class="n">Seabold</span><span class="p">,</span> <span class="n">Skipper</span> <span class="ow">and</span> <span class="n">Perktold</span><span class="p">,</span> <span class="n">Josef</span><span class="p">},</span>
<span class="n">booktitle</span><span class="o">=</span><span class="p">{</span><span class="mi">9</span><span class="n">th</span> <span class="n">Python</span> <span class="ow">in</span> <span class="n">Science</span> <span class="n">Conference</span><span class="p">},</span>
<span class="n">year</span><span class="o">=</span><span class="p">{</span><span class="mi">2010</span><span class="p">},</span>
<span class="p">}</span>
</code></pre></div>
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