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Fitting models using R-style formulas
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<h1 id="fitting-models-using-r-style-formulas"><span id="formula-examples"></span>Fitting models using R-style formulas<a class="headerlink" href="#fitting-models-using-r-style-formulas" title="Link to this heading">¶</a></h1>
<p>Since version 0.5.0, <code class="docutils literal notranslate"><span class="pre">statsmodels</span></code> allows users to fit statistical
models using R-style formulas. Internally, <code class="docutils literal notranslate"><span class="pre">statsmodels</span></code> uses the
<a class="reference external" href="https://patsy.readthedocs.io/en/latest/">patsy</a> package to convert formulas and
data to the matrices that are used in model fitting. The formula
framework is quite powerful; this tutorial only scratches the surface. A
full description of the formula language can be found in the <code class="docutils literal notranslate"><span class="pre">patsy</span></code>
docs:</p>
<ul class="simple">
<li><p><a class="reference external" href="https://patsy.readthedocs.io/en/latest/">Patsy formula language description</a></p></li>
</ul>
<h2 id="loading-modules-and-functions">Loading modules and functions<a class="headerlink" href="#loading-modules-and-functions" title="Link to this heading">¶</a></h2>
<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">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 [2]: </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="gp">In [3]: </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 [4]: </span><span class="kn">import</span><span class="w"> </span><span class="nn">pandas</span>
</code></pre></div>
</div>
<p>Notice that we called <code class="docutils literal notranslate"><span class="pre">statsmodels.formula.api</span></code> in addition to the usual
<code class="docutils literal notranslate"><span class="pre">statsmodels.api</span></code>. In fact, <code class="docutils literal notranslate"><span class="pre">statsmodels.api</span></code> is used here only to load
the dataset. The <code class="docutils literal notranslate"><span class="pre">formula.api</span></code> hosts many of the same
functions found in <code class="docutils literal notranslate"><span class="pre">api</span></code> (e.g. OLS, GLM), but it also holds lower case
counterparts for most of these models. In general, lower case models
accept <code class="docutils literal notranslate"><span class="pre">formula</span></code> and <code class="docutils literal notranslate"><span class="pre">df</span></code> arguments, whereas upper case ones take
<code class="docutils literal notranslate"><span class="pre">endog</span></code> and <code class="docutils literal notranslate"><span class="pre">exog</span></code> design matrices. <code class="docutils literal notranslate"><span class="pre">formula</span></code> accepts a string
which describes the model in terms of a <code class="docutils literal notranslate"><span class="pre">patsy</span></code> formula. <code class="docutils literal notranslate"><span class="pre">df</span></code> takes
a <a class="reference external" href="https://pandas.pydata.org/">pandas</a> data frame.</p>
<p><code class="docutils literal notranslate"><span class="pre">dir(smf)</span></code> will print a list of available models.</p>
<p>Formula-compatible models have the following generic call signature:
<code class="docutils literal notranslate"><span class="pre">(formula,</span> <span class="pre">data,</span> <span class="pre">subset=None,</span> <span class="pre">*args,</span> <span class="pre">**kwargs)</span></code></p>
<h2 id="ols-regression-using-formulas">OLS regression using formulas<a class="headerlink" href="#ols-regression-using-formulas" title="Link to this heading">¶</a></h2>
<p>To begin, we fit the linear model described on the <a class="reference external" href="gettingstarted.html">Getting
Started</a> page. Download the data, subset columns,
and list-wise delete to remove missing observations:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [5]: </span><span class="n">df</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="gp">In [6]: </span><span class="n">df</span> <span class="o">=</span> <span class="n">df</span><span class="p">[[</span><span class="s1">'Lottery'</span><span class="p">,</span> <span class="s1">'Literacy'</span><span class="p">,</span> <span class="s1">'Wealth'</span><span class="p">,</span> <span class="s1">'Region'</span><span class="p">]]</span><span class="o">.</span><span class="n">dropna</span><span class="p">()</span>
<span class="gp">In [7]: </span><span class="n">df</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
<span class="gh">Out[7]: </span>
<span class="go"> Lottery Literacy Wealth Region</span>
<span class="go">0 41 37 73 E</span>
<span class="go">1 38 51 22 N</span>
<span class="go">2 66 13 61 C</span>
<span class="go">3 80 46 76 E</span>
<span class="go">4 79 69 83 E</span>
</code></pre></div>
</div>
<p>Fit the model:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [8]: </span><span class="n">mod</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="n">formula</span><span class="o">=</span><span class="s1">'Lottery ~ Literacy + Wealth + Region'</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df</span><span class="p">)</span>
<span class="gp">In [9]: </span><span class="n">res</span> <span class="o">=</span> <span class="n">mod</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="gp">In [10]: </span><span class="nb">print</span><span class="p">(</span><span class="n">res</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: 0.338</span>
<span class="go">Model: OLS Adj. R-squared: 0.287</span>
<span class="go">Method: Least Squares F-statistic: 6.636</span>
<span class="go">Date: Wed, 19 Feb 2025 Prob (F-statistic): 1.07e-05</span>
<span class="go">Time: 12:52:17 Log-Likelihood: -375.30</span>
<span class="go">No. Observations: 85 AIC: 764.6</span>
<span class="go">Df Residuals: 78 BIC: 781.7</span>
<span class="go">Df Model: 6 </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">38.6517</span> <span class="mf">9.456</span> <span class="mf">4.087</span> <span class="mf">0.000</span> <span class="mf">19.826</span> <span class="mf">57.478</span>
<span class="n">Region</span><span class="p">[</span><span class="n">T</span><span class="o">.</span><span class="n">E</span><span class="p">]</span> <span class="o">-</span><span class="mf">15.4278</span> <span class="mf">9.727</span> <span class="o">-</span><span class="mf">1.586</span> <span class="mf">0.117</span> <span class="o">-</span><span class="mf">34.793</span> <span class="mf">3.938</span>
<span class="n">Region</span><span class="p">[</span><span class="n">T</span><span class="o">.</span><span class="n">N</span><span class="p">]</span> <span class="o">-</span><span class="mf">10.0170</span> <span class="mf">9.260</span> <span class="o">-</span><span class="mf">1.082</span> <span class="mf">0.283</span> <span class="o">-</span><span class="mf">28.453</span> <span class="mf">8.419</span>
<span class="n">Region</span><span class="p">[</span><span class="n">T</span><span class="o">.</span><span class="n">S</span><span class="p">]</span> <span class="o">-</span><span class="mf">4.5483</span> <span class="mf">7.279</span> <span class="o">-</span><span class="mf">0.625</span> <span class="mf">0.534</span> <span class="o">-</span><span class="mf">19.039</span> <span class="mf">9.943</span>
<span class="n">Region</span><span class="p">[</span><span class="n">T</span><span class="o">.</span><span class="n">W</span><span class="p">]</span> <span class="o">-</span><span class="mf">10.0913</span> <span class="mf">7.196</span> <span class="o">-</span><span class="mf">1.402</span> <span class="mf">0.165</span> <span class="o">-</span><span class="mf">24.418</span> <span class="mf">4.235</span>
<span class="n">Literacy</span> <span class="o">-</span><span class="mf">0.1858</span> <span class="mf">0.210</span> <span class="o">-</span><span class="mf">0.886</span> <span class="mf">0.378</span> <span class="o">-</span><span class="mf">0.603</span> <span class="mf">0.232</span>
<span class="n">Wealth</span> <span class="mf">0.4515</span> <span class="mf">0.103</span> <span class="mf">4.390</span> <span class="mf">0.000</span> <span class="mf">0.247</span> <span class="mf">0.656</span>
<span class="o">==============================================================================</span>
<span class="ne">Omnibus</span>: 3.049 Durbin-Watson: 1.785
<span class="n">Prob</span><span class="p">(</span><span class="n">Omnibus</span><span class="p">):</span> <span class="mf">0.218</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">2.694</span>
<span class="ne">Skew</span>: -0.340 Prob(JB): 0.260
<span class="ne">Kurtosis</span>: 2.454 Cond. No. 371.
<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>
<h2 id="categorical-variables">Categorical variables<a class="headerlink" href="#categorical-variables" title="Link to this heading">¶</a></h2>
<p>Looking at the summary printed above, notice that <code class="docutils literal notranslate"><span class="pre">patsy</span></code> determined
that elements of <em>Region</em> were text strings, so it treated <em>Region</em> as a
categorical variable. <code class="docutils literal notranslate"><span class="pre">patsy</span></code>’s default is also to include an
intercept, so we automatically dropped one of the <em>Region</em> categories.</p>
<p>If <em>Region</em> had been an integer variable that we wanted to treat
explicitly as categorical, we could have done so by using the <code class="docutils literal notranslate"><span class="pre">C()</span></code>
operator:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [11]: </span><span class="n">res</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="n">formula</span><span class="o">=</span><span class="s1">'Lottery ~ Literacy + Wealth + C(Region)'</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="gp">In [12]: </span><span class="nb">print</span><span class="p">(</span><span class="n">res</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
<span class="go">Intercept 38.651655</span>
<span class="go">C(Region)[T.E] -15.427785</span>
<span class="go">C(Region)[T.N] -10.016961</span>
<span class="go">C(Region)[T.S] -4.548257</span>
<span class="go">C(Region)[T.W] -10.091276</span>
<span class="go">Literacy -0.185819</span>
<span class="go">Wealth 0.451475</span>
<span class="go">dtype: float64</span>
</code></pre></div>
</div>
<p>Examples more advanced features <code class="docutils literal notranslate"><span class="pre">patsy</span></code>’s categorical variables
function can be found here: <a class="reference external" href="contrasts.html">Patsy: Contrast Coding Systems for
categorical variables</a></p>
<h2 id="operators">Operators<a class="headerlink" href="#operators" title="Link to this heading">¶</a></h2>
<p>We have already seen that “~” separates the left-hand side of the model
from the right-hand side, and that “+” adds new columns to the design
matrix.</p>
<h3 id="removing-variables">Removing variables<a class="headerlink" href="#removing-variables" title="Link to this heading">¶</a></h3>
<p>The “-” sign can be used to remove columns/variables. For instance, we
can remove the intercept from a model by:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [13]: </span><span class="n">res</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="n">formula</span><span class="o">=</span><span class="s1">'Lottery ~ Literacy + Wealth + C(Region) -1 '</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="gp">In [14]: </span><span class="nb">print</span><span class="p">(</span><span class="n">res</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
<span class="go">C(Region)[C] 38.651655</span>
<span class="go">C(Region)[E] 23.223870</span>
<span class="go">C(Region)[N] 28.634694</span>
<span class="go">C(Region)[S] 34.103399</span>
<span class="go">C(Region)[W] 28.560379</span>
<span class="go">Literacy -0.185819</span>
<span class="go">Wealth 0.451475</span>
<span class="go">dtype: float64</span>
</code></pre></div>
</div>
<h3 id="multiplicative-interactions">Multiplicative interactions<a class="headerlink" href="#multiplicative-interactions" title="Link to this heading">¶</a></h3>
<p>“:” adds a new column to the design matrix with the product of the other
two columns. “*” will also include the individual columns that were
multiplied together:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [15]: </span><span class="n">res1</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="n">formula</span><span class="o">=</span><span class="s1">'Lottery ~ Literacy : Wealth - 1'</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="gp">In [16]: </span><span class="n">res2</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="n">formula</span><span class="o">=</span><span class="s1">'Lottery ~ Literacy * Wealth - 1'</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="gp">In [17]: </span><span class="nb">print</span><span class="p">(</span><span class="n">res1</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
<span class="go">Literacy:Wealth 0.018176</span>
<span class="go">dtype: float64</span>
<span class="gp">In [18]: </span><span class="nb">print</span><span class="p">(</span><span class="n">res2</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
<span class="go">Literacy 0.427386</span>
<span class="go">Wealth 1.080987</span>
<span class="go">Literacy:Wealth -0.013609</span>
<span class="go">dtype: float64</span>
</code></pre></div>
</div>
<p>Many other things are possible with operators. Please consult the <a class="reference external" href="https://patsy.readthedocs.io/en/latest/formulas.html">patsy
docs</a> to learn
more.</p>
<h2 id="functions">Functions<a class="headerlink" href="#functions" title="Link to this heading">¶</a></h2>
<p>You can apply vectorized functions to the variables in your model:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [19]: </span><span class="n">res</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="n">formula</span><span class="o">=</span><span class="s1">'Lottery ~ np.log(Literacy)'</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="gp">In [20]: </span><span class="nb">print</span><span class="p">(</span><span class="n">res</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
<span class="go">Intercept 115.609119</span>
<span class="go">np.log(Literacy) -20.393959</span>
<span class="go">dtype: float64</span>
</code></pre></div>
</div>
<p>Define a custom function:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [21]: </span><span class="k">def</span><span class="w"> </span><span class="nf">log_plus_1</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="gp"> ....: </span> <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1.0</span>
<span class="gp"> ....: </span>
<span class="gp">In [22]: </span><span class="n">res</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="n">formula</span><span class="o">=</span><span class="s1">'Lottery ~ log_plus_1(Literacy)'</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="gp">In [23]: </span><span class="nb">print</span><span class="p">(</span><span class="n">res</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
<span class="go">Intercept 136.003079</span>
<span class="go">log_plus_1(Literacy) -20.393959</span>
<span class="go">dtype: float64</span>
</code></pre></div>
</div>
<h2 id="namespaces"><span id="patsy-namespaces"></span>Namespaces<a class="headerlink" href="#namespaces" title="Link to this heading">¶</a></h2>
<p>Notice that all of the above examples use the calling namespace to look for the functions to apply. The namespace used can be controlled via the <code class="docutils literal notranslate"><span class="pre">eval_env</span></code> keyword. For example, you may want to give a custom namespace using the <code class="xref py py-class docutils literal notranslate"><span class="pre">patsy:patsy.EvalEnvironment</span></code> or you may want to use a “clean” namespace, which we provide by passing <code class="docutils literal notranslate"><span class="pre">eval_func=-1</span></code>. The default is to use the caller’s namespace. This can have (un)expected consequences, if, for example, someone has a variable names <code class="docutils literal notranslate"><span class="pre">C</span></code> in the user namespace or in their data structure passed to <code class="docutils literal notranslate"><span class="pre">patsy</span></code>, and <code class="docutils literal notranslate"><span class="pre">C</span></code> is used in the formula to handle a categorical variable. See the <a class="reference external" href="https://patsy.readthedocs.io/en/latest/API-reference.html">Patsy API Reference</a> for more information.</p>
<h2 id="using-formulas-with-models-that-do-not-yet-support-them">Using formulas with models that do not (yet) support them<a class="headerlink" href="#using-formulas-with-models-that-do-not-yet-support-them" title="Link to this heading">¶</a></h2>
<p>Even if a given <code class="docutils literal notranslate"><span class="pre">statsmodels</span></code> function does not support formulas, you
can still use <code class="docutils literal notranslate"><span class="pre">patsy</span></code>’s formula language to produce design matrices.
Those matrices can then be fed to the fitting function as <code class="docutils literal notranslate"><span class="pre">endog</span></code> and
<code class="docutils literal notranslate"><span class="pre">exog</span></code> arguments.</p>
<p>To generate <code class="docutils literal notranslate"><span class="pre">numpy</span></code> arrays:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [24]: </span><span class="kn">import</span><span class="w"> </span><span class="nn">patsy</span>
<span class="gp">In [25]: </span><span class="n">f</span> <span class="o">=</span> <span class="s1">'Lottery ~ Literacy * Wealth'</span>
<span class="gp">In [26]: </span><span class="n">y</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="n">patsy</span><span class="o">.</span><span class="n">dmatrices</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">df</span><span class="p">,</span> <span class="n">return_type</span><span class="o">=</span><span class="s1">'matrix'</span><span class="p">)</span>
<span class="gp">In [27]: </span><span class="nb">print</span><span class="p">(</span><span class="n">y</span><span class="p">[:</span><span class="mi">5</span><span class="p">])</span>
<span class="go">[[41.]</span>
<span class="go"> [38.]</span>
<span class="go"> [66.]</span>
<span class="go"> [80.]</span>
<span class="go"> [79.]]</span>
<span class="gp">In [28]: </span><span class="nb">print</span><span class="p">(</span><span class="n">X</span><span class="p">[:</span><span class="mi">5</span><span class="p">])</span>
<span class="go">[[ 1. 37. 73. 2701.]</span>
<span class="go"> [ 1. 51. 22. 1122.]</span>
<span class="go"> [ 1. 13. 61. 793.]</span>
<span class="go"> [ 1. 46. 76. 3496.]</span>
<span class="go"> [ 1. 69. 83. 5727.]]</span>
</code></pre></div>
</div>
<p><code class="docutils literal notranslate"><span class="pre">y</span></code> and <code class="docutils literal notranslate"><span class="pre">X</span></code> would be instances of <code class="docutils literal notranslate"><span class="pre">patsy.DesignMatrix</span></code> which is a subclass of <code class="docutils literal notranslate"><span class="pre">numpy.ndarray</span></code>.</p>
<p>To generate pandas data frames:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [29]: </span><span class="n">f</span> <span class="o">=</span> <span class="s1">'Lottery ~ Literacy * Wealth'</span>
<span class="gp">In [30]: </span><span class="n">y</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="n">patsy</span><span class="o">.</span><span class="n">dmatrices</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">df</span><span class="p">,</span> <span class="n">return_type</span><span class="o">=</span><span class="s1">'dataframe'</span><span class="p">)</span>
<span class="gp">In [31]: </span><span class="nb">print</span><span class="p">(</span><span class="n">y</span><span class="p">[:</span><span class="mi">5</span><span class="p">])</span>
<span class="go"> Lottery</span>
<span class="go">0 41.0</span>
<span class="go">1 38.0</span>
<span class="go">2 66.0</span>
<span class="go">3 80.0</span>
<span class="go">4 79.0</span>
<span class="gp">In [32]: </span><span class="nb">print</span><span class="p">(</span><span class="n">X</span><span class="p">[:</span><span class="mi">5</span><span class="p">])</span>
<span class="go"> Intercept Literacy Wealth Literacy:Wealth</span>
<span class="go">0 1.0 37.0 73.0 2701.0</span>
<span class="go">1 1.0 51.0 22.0 1122.0</span>
<span class="go">2 1.0 13.0 61.0 793.0</span>
<span class="go">3 1.0 46.0 76.0 3496.0</span>
<span class="go">4 1.0 69.0 83.0 5727.0</span>
</code></pre></div>
</div>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [33]: </span><span class="nb">print</span><span class="p">(</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="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: 0.309</span>
<span class="go">Model: OLS Adj. R-squared: 0.283</span>
<span class="go">Method: Least Squares F-statistic: 12.06</span>