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gettingstarted.html
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<!doctype html>
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<title>Getting started - statsmodels 0.15.0 (+647)</title>
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statsmodels 0.15.0 (+647)
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Getting started
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<h1 id="getting-started">Getting started<a class="headerlink" href="#getting-started" title="Link to this heading">¶</a></h1>
<p>This very simple case-study is designed to get you up-and-running quickly with
<code class="docutils literal notranslate"><span class="pre">statsmodels</span></code>. Starting from raw data, we will show the steps needed to
estimate a statistical model and to draw a diagnostic plot. We will only use
functions provided by <code class="docutils literal notranslate"><span class="pre">statsmodels</span></code> or its <code class="docutils literal notranslate"><span class="pre">pandas</span></code> and <code class="docutils literal notranslate"><span class="pre">patsy</span></code>
dependencies.</p>
<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>
<p>After <a class="reference external" href="install.html">installing statsmodels and its dependencies</a>, we load a
few modules and functions:</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">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">pandas</span>
<span class="gp">In [3]: </span><span class="kn">from</span><span class="w"> </span><span class="nn">patsy</span><span class="w"> </span><span class="kn">import</span> <span class="n">dmatrices</span>
</code></pre></div>
</div>
<p><a class="reference external" href="https://pandas.pydata.org/">pandas</a> builds on <code class="docutils literal notranslate"><span class="pre">numpy</span></code> arrays to provide
rich data structures and data analysis tools. The <code class="docutils literal notranslate"><span class="pre">pandas.DataFrame</span></code> function
provides labelled arrays of (potentially heterogenous) data, similar to the
<code class="docutils literal notranslate"><span class="pre">R</span></code> “data.frame”. The <code class="docutils literal notranslate"><span class="pre">pandas.read_csv</span></code> function can be used to convert a
comma-separated values file to a <code class="docutils literal notranslate"><span class="pre">DataFrame</span></code> object.</p>
<p><a class="reference external" href="https://github.com/pydata/patsy">patsy</a> is a Python library for describing
statistical models and building <a class="reference external" href="https://en.wikipedia.org/wiki/Design_matrix">Design Matrices</a> using <code class="docutils literal notranslate"><span class="pre">R</span></code>-like formulas.</p>
<div class="note admonition">
<p class="admonition-title">Note</p>
<p>This example uses the API interface. See <a class="reference internal" href="api-structure.html#importpaths"><span class="std std-ref">Import Paths and Structure</span></a> for information on
the difference between importing the API interfaces (<code class="docutils literal notranslate"><span class="pre">statsmodels.api</span></code> and
<code class="docutils literal notranslate"><span class="pre">statsmodels.tsa.api</span></code>) and directly importing from the module that defines
the model.</p>
</div>
<h2 id="data">Data<a class="headerlink" href="#data" title="Link to this heading">¶</a></h2>
<p>We download the <a class="reference external" href="https://vincentarelbundock.github.io/Rdatasets/doc/HistData/Guerry.html">Guerry dataset</a>, a
collection of historical data used in support of Andre-Michel Guerry’s 1833
<em>Essay on the Moral Statistics of France</em>. The data set is hosted online in
comma-separated values format (CSV) by the <a class="reference external" href="https://github.com/vincentarelbundock/Rdatasets/">Rdatasets</a> repository.
We could download the file locally and then load it using <code class="docutils literal notranslate"><span class="pre">read_csv</span></code>, but
<code class="docutils literal notranslate"><span class="pre">pandas</span></code> takes care of all of this automatically for us:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [4]: </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>
</code></pre></div>
</div>
<p>The <a class="reference external" href="iolib.html">Input/Output doc page</a> shows how to import from various
other formats.</p>
<p>We select the variables of interest and look at the bottom 5 rows:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [5]: </span><span class="nb">vars</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'Department'</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="gp">In [6]: </span><span class="n">df</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="nb">vars</span><span class="p">]</span>
<span class="gp">In [7]: </span><span class="n">df</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">:]</span>
<span class="gh">Out[7]: </span>
<span class="go"> Department Lottery Literacy Wealth Region</span>
<span class="go">81 Vienne 40 25 68 W</span>
<span class="go">82 Haute-Vienne 55 13 67 C</span>
<span class="go">83 Vosges 14 62 82 E</span>
<span class="go">84 Yonne 51 47 30 C</span>
<span class="go">85 Corse 83 49 37 NaN</span>
</code></pre></div>
</div>
<p>Notice that there is one missing observation in the <em>Region</em> column. We
eliminate it using a <code class="docutils literal notranslate"><span class="pre">DataFrame</span></code> method provided by <code class="docutils literal notranslate"><span class="pre">pandas</span></code>:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [8]: </span><span class="n">df</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">dropna</span><span class="p">()</span>
<span class="gp">In [9]: </span><span class="n">df</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">:]</span>
<span class="gh">Out[9]: </span>
<span class="go"> Department Lottery Literacy Wealth Region</span>
<span class="go">80 Vendee 68 28 56 W</span>
<span class="go">81 Vienne 40 25 68 W</span>
<span class="go">82 Haute-Vienne 55 13 67 C</span>
<span class="go">83 Vosges 14 62 82 E</span>
<span class="go">84 Yonne 51 47 30 C</span>
</code></pre></div>
</div>
<h2 id="substantive-motivation-and-model">Substantive motivation and model<a class="headerlink" href="#substantive-motivation-and-model" title="Link to this heading">¶</a></h2>
<p>We want to know whether literacy rates in the 86 French departments are
associated with per capita wagers on the Royal Lottery in the 1820s. We need to
control for the level of wealth in each department, and we also want to include
a series of dummy variables on the right-hand side of our regression equation to
control for unobserved heterogeneity due to regional effects. The model is
estimated using ordinary least squares regression (OLS).</p>
<h2 id="design-matrices-endog-exog">Design matrices (<em>endog</em> & <em>exog</em>)<a class="headerlink" href="#design-matrices-endog-exog" title="Link to this heading">¶</a></h2>
<p>To fit most of the models covered by <code class="docutils literal notranslate"><span class="pre">statsmodels</span></code>, you will need to create
two design matrices. The first is a matrix of endogenous variable(s) (i.e.
dependent, response, regressand, etc.). The second is a matrix of exogenous
variable(s) (i.e. independent, predictor, regressor, etc.). The OLS coefficient
estimates are calculated as usual:</p>
<div class="math notranslate nohighlight">
\[\hat{\beta} = (X'X)^{-1} X'y\]</div>
<p>where <span class="math notranslate nohighlight">\(y\)</span> is an <span class="math notranslate nohighlight">\(N \times 1\)</span> column of data on lottery wagers per
capita (<em>Lottery</em>). <span class="math notranslate nohighlight">\(X\)</span> is <span class="math notranslate nohighlight">\(N \times 7\)</span> with an intercept, the
<em>Literacy</em> and <em>Wealth</em> variables, and 4 region binary variables.</p>
<p>The <code class="docutils literal notranslate"><span class="pre">patsy</span></code> module provides a convenient function to prepare design matrices
using <code class="docutils literal notranslate"><span class="pre">R</span></code>-like formulas. You can find more information <a class="reference external" href="https://patsy.readthedocs.io/en/latest/">here</a>.</p>
<p>We use <code class="docutils literal notranslate"><span class="pre">patsy</span></code>’s <code class="docutils literal notranslate"><span class="pre">dmatrices</span></code> function to create design matrices:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [10]: </span><span class="n">y</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="n">dmatrices</span><span class="p">(</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="n">return_type</span><span class="o">=</span><span class="s1">'dataframe'</span><span class="p">)</span>
</code></pre></div>
</div>
<p>The resulting matrices/data frames look like this:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [11]: </span><span class="n">y</span><span class="p">[:</span><span class="mi">3</span><span class="p">]</span>
<span class="gh">Out[11]: </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="gp">In [12]: </span><span class="n">X</span><span class="p">[:</span><span class="mi">3</span><span class="p">]</span>
<span class="gh">Out[12]: </span>
<span class="go"> Intercept Region[T.E] Region[T.N] ... Region[T.W] Literacy Wealth</span>
<span class="go">0 1.0 1.0 0.0 ... 0.0 37.0 73.0</span>
<span class="go">1 1.0 0.0 1.0 ... 0.0 51.0 22.0</span>
<span class="go">2 1.0 0.0 0.0 ... 0.0 13.0 61.0</span>
<span class="go">[3 rows x 7 columns]</span>
</code></pre></div>
</div>
<p>Notice that <code class="docutils literal notranslate"><span class="pre">dmatrices</span></code> has</p>
<ul class="simple">
<li><p>split the categorical <em>Region</em> variable into a set of indicator variables.</p></li>
<li><p>added a constant to the exogenous regressors matrix.</p></li>
<li><p>returned <code class="docutils literal notranslate"><span class="pre">pandas</span></code> DataFrames instead of simple numpy arrays. This is useful because DataFrames allow <code class="docutils literal notranslate"><span class="pre">statsmodels</span></code> to carry-over meta-data (e.g. variable names) when reporting results.</p></li>
</ul>
<p>The above behavior can of course be altered. See the <a class="reference external" href="https://patsy.readthedocs.io/en/latest/">patsy doc pages</a>.</p>
<h2 id="model-fit-and-summary">Model fit and summary<a class="headerlink" href="#model-fit-and-summary" title="Link to this heading">¶</a></h2>
<p>Fitting a model in <code class="docutils literal notranslate"><span class="pre">statsmodels</span></code> typically involves 3 easy steps:</p>
<ol class="arabic simple">
<li><p>Use the model class to describe the model</p></li>
<li><p>Fit the model using a class method</p></li>
<li><p>Inspect the results using a summary method</p></li>
</ol>
<p>For OLS, this is achieved by:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [13]: </span><span class="n">mod</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="c1"># Describe model</span>
<span class="gp">In [14]: </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="c1"># Fit model</span>
<span class="gp">In [15]: </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="c1"># Summarize model</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, 02 Apr 2025 Prob (F-statistic): 1.07e-05</span>
<span class="go">Time: 08:47:09 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>
<p>The <code class="docutils literal notranslate"><span class="pre">res</span></code> object has many useful attributes. For example, we can extract
parameter estimates and r-squared by typing:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [16]: </span><span class="n">res</span><span class="o">.</span><span class="n">params</span>
<span class="gh">Out[16]: </span>
<span class="go">Intercept 38.651655</span>
<span class="go">Region[T.E] -15.427785</span>
<span class="go">Region[T.N] -10.016961</span>
<span class="go">Region[T.S] -4.548257</span>
<span class="go">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>
<span class="gp">In [17]: </span><span class="n">res</span><span class="o">.</span><span class="n">rsquared</span>
<span class="gh">Out[17]: </span><span class="go">np.float64(0.337950869192882)</span>
</code></pre></div>
</div>
<p>Type <code class="docutils literal notranslate"><span class="pre">dir(res)</span></code> for a full list of attributes.</p>
<p>For more information and examples, see the <a class="reference external" href="regression.html">Regression doc page</a></p>
<h2 id="diagnostics-and-specification-tests">Diagnostics and specification tests<a class="headerlink" href="#diagnostics-and-specification-tests" title="Link to this heading">¶</a></h2>
<p><code class="docutils literal notranslate"><span class="pre">statsmodels</span></code> allows you to conduct a range of useful <a class="reference external" href="stats.html#residual-diagnostics-and-specification-tests">regression diagnostics
and specification tests</a>. For instance,
apply the Rainbow test for linearity (the null hypothesis is that the
relationship is properly modelled as linear):</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [18]: </span><span class="n">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">linear_rainbow</span><span class="p">(</span><span class="n">res</span><span class="p">)</span>
<span class="gh">Out[18]: </span><span class="go">(np.float64(0.847233997615691), np.float64(0.6997965543621644))</span>
</code></pre></div>
</div>
<p>Admittedly, the output produced above is not very verbose, but we know from
reading the <a class="reference external" href="generated/statsmodels.stats.diagnostic.linear_rainbow.html">docstring</a>
(also, <code class="docutils literal notranslate"><span class="pre">print(sm.stats.linear_rainbow.__doc__)</span></code>) that the
first number is an F-statistic and that the second is the p-value.</p>
<p><code class="docutils literal notranslate"><span class="pre">statsmodels</span></code> also provides graphics functions. For example, we can draw a
plot of partial regression for a set of regressors by:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [19]: </span><span class="n">sm</span><span class="o">.</span><span class="n">graphics</span><span class="o">.</span><span class="n">plot_partregress</span><span class="p">(</span><span class="s1">'Lottery'</span><span class="p">,</span> <span class="s1">'Wealth'</span><span class="p">,</span> <span class="p">[</span><span class="s1">'Region'</span><span class="p">,</span> <span class="s1">'Literacy'</span><span class="p">],</span>
<span class="gp"> ....: </span> <span class="n">data</span><span class="o">=</span><span class="n">df</span><span class="p">,</span> <span class="n">obs_labels</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="gp"> ....: </span>
<span class="gh">Out[19]: </span><span class="go"><Figure size 640x480 with 1 Axes></span>
</code></pre></div>
</div>
<img alt="_images/gettingstarted_0.png" src="_images/gettingstarted_0.png" />
<h2 id="documentation">Documentation<a class="headerlink" href="#documentation" title="Link to this heading">¶</a></h2>
<p>Documentation can be accessed from an IPython session
using <a class="reference internal" href="generated/statsmodels.tools.web.webdoc.html#statsmodels.tools.web.webdoc" title="statsmodels.tools.web.webdoc (Python function) — Opens a browser and displays online documentation"><code class="xref py py-func docutils literal notranslate"><span class="pre">webdoc</span></code></a>.</p>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tools.web.webdoc.html#statsmodels.tools.web.webdoc" title="statsmodels.tools.web.webdoc (Python function) — Opens a browser and displays online documentation"><code class="xref py py-obj docutils literal notranslate"><span class="pre">webdoc</span></code></a></p></td>