Quantile regression
===================


.. _quantile_regression_notebook:

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

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   <p>This example page shows how to use <code>statsmodels</code>&#39; <code>QuantReg</code> class to replicate parts of the analysis published in </p>
   <ul>
   <li>Koenker, Roger and Kevin F. Hallock. &quot;Quantile Regressioin&quot;. Journal of Economic Perspectives, Volume 15, Number 4, Fall 2001, Pages 143–156</li>
   </ul>
   <p>We are interested in the relationship between income and expenditures on food for a sample of working class Belgian households in 1857 (the Engel data). </p>
   <h2 id="setup">Setup</h2>
   <p>We first need to load some modules and to retrieve the data. Conveniently, the Engel dataset is shipped with <code>statsmodels</code>.</p>
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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">import</span> <span class="nn">patsy</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.api</span> <span class="kn">as</span> <span class="nn">sm</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">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
   <span class="kn">from</span> <span class="nn">statsmodels.regression.quantile_regression</span> <span class="kn">import</span> <span class="n">QuantReg</span>
   
   <span class="n">data</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">engel</span><span class="o">.</span><span class="n">load_pandas</span><span class="p">()</span><span class="o">.</span><span class="n">data</span>
   <span class="n">data</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
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   <table border="1" class="dataframe">
     <thead>
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         <th></th>
         <th>income</th>
         <th>foodexp</th>
       </tr>
     </thead>
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       <tr>
         <th>0</th>
         <td>420.157651</td>
         <td>255.839425</td>
       </tr>
       <tr>
         <th>1</th>
         <td>541.411707</td>
         <td>310.958667</td>
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         <th>2</th>
         <td>901.157457</td>
         <td>485.680014</td>
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         <th>3</th>
         <td>639.080229</td>
         <td>402.997356</td>
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         <th>4</th>
         <td>750.875606</td>
         <td>495.560775</td>
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   <h2 id="least-absolute-deviation">Least Absolute Deviation</h2>
   <p>The LAD model is a special case of quantile regression where q=0.5</p>
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   <div class="highlight"><pre><span class="n">mod</span> <span class="o">=</span> <span class="n">smf</span><span class="o">.</span><span class="n">quantreg</span><span class="p">(</span><span class="s1">&#39;foodexp ~ income&#39;</span><span class="p">,</span> <span class="n">data</span><span class="p">)</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="n">q</span><span class="o">=.</span><span class="mi">5</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">res</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
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                            QuantReg Regression Results                          
   ==============================================================================
   Dep. Variable:                foodexp   Pseudo R-squared:               0.6206
   Model:                       QuantReg   Bandwidth:                       64.51
   Method:                 Least Squares   Sparsity:                        209.3
   Date:                Wed, 27 Apr 2016   No. Observations:                  235
   Time:                        23:31:31   Df Residuals:                      233
                                           Df Model:                            1
   ==============================================================================
                    coef    std err          t      P&gt;|t|      [95.0% Conf. Int.]
   ------------------------------------------------------------------------------
   Intercept     81.4823     14.634      5.568      0.000        52.649   110.315
   income         0.5602      0.013     42.516      0.000         0.534     0.586
   ==============================================================================
   
   The condition number is large, 2.38e+03. This might indicate that there are
   strong multicollinearity or other numerical problems.
   
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   <h2 id="visualizing-the-results">Visualizing the results</h2>
   <p>We estimate the quantile regression model for many quantiles between .05 and .95, and compare best fit line from each of these models to Ordinary Least Squares results. </p>
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   <h3 id="prepare-data-for-plotting">Prepare data for plotting</h3>
   <p>For convenience, we place the quantile regression results in a Pandas DataFrame, and the OLS results in a dictionary.</p>
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   <div class="highlight"><pre><span class="n">quantiles</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">.</span><span class="mo">05</span><span class="p">,</span> <span class="o">.</span><span class="mi">96</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">)</span>
   <span class="k">def</span> <span class="nf">fit_model</span><span class="p">(</span><span class="n">q</span><span class="p">):</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="n">q</span><span class="o">=</span><span class="n">q</span><span class="p">)</span>
       <span class="k">return</span> <span class="p">[</span><span class="n">q</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="s1">&#39;Intercept&#39;</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="s1">&#39;income&#39;</span><span class="p">]]</span> <span class="o">+</span> \
               <span class="n">res</span><span class="o">.</span><span class="n">conf_int</span><span class="p">()</span><span class="o">.</span><span class="n">ix</span><span class="p">[</span><span class="s1">&#39;income&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
       
   <span class="n">models</span> <span class="o">=</span> <span class="p">[</span><span class="n">fit_model</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">quantiles</span><span class="p">]</span>
   <span class="n">models</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">models</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;q&#39;</span><span class="p">,</span> <span class="s1">&#39;a&#39;</span><span class="p">,</span> <span class="s1">&#39;b&#39;</span><span class="p">,</span><span class="s1">&#39;lb&#39;</span><span class="p">,</span><span class="s1">&#39;ub&#39;</span><span class="p">])</span>
   
   <span class="n">ols</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">&#39;foodexp ~ income&#39;</span><span class="p">,</span> <span class="n">data</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
   <span class="n">ols_ci</span> <span class="o">=</span> <span class="n">ols</span><span class="o">.</span><span class="n">conf_int</span><span class="p">()</span><span class="o">.</span><span class="n">ix</span><span class="p">[</span><span class="s1">&#39;income&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
   <span class="n">ols</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">a</span> <span class="o">=</span> <span class="n">ols</span><span class="o">.</span><span class="n">params</span><span class="p">[</span><span class="s1">&#39;Intercept&#39;</span><span class="p">],</span>
              <span class="n">b</span> <span class="o">=</span> <span class="n">ols</span><span class="o">.</span><span class="n">params</span><span class="p">[</span><span class="s1">&#39;income&#39;</span><span class="p">],</span>
              <span class="n">lb</span> <span class="o">=</span> <span class="n">ols_ci</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
              <span class="n">ub</span> <span class="o">=</span> <span class="n">ols_ci</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
   
   <span class="k">print</span><span class="p">(</span><span class="n">models</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">ols</span><span class="p">)</span>
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         q           a         b        lb        ub
   0  0.05  124.880099  0.343361  0.268632  0.418090
   1  0.15  111.693660  0.423708  0.382780  0.464636
   2  0.25   95.483539  0.474103  0.439900  0.508306
   3  0.35  105.841294  0.488901  0.457759  0.520043
   4  0.45   81.083647  0.552428  0.525021  0.579835
   5  0.55   89.661370  0.565601  0.540955  0.590247
   6  0.65   74.033434  0.604576  0.582169  0.626982
   7  0.75   62.396584  0.644014  0.622411  0.665617
   8  0.85   52.272216  0.677603  0.657383  0.697823
   9  0.95   64.103964  0.709069  0.687831  0.730306
   {&apos;a&apos;: 147.47538852370633, &apos;b&apos;: 0.48517842367692343, &apos;lb&apos;: 0.45687381301842317, &apos;ub&apos;: 0.51348303433542364}
   
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   /build/statsmodels-ungkPp/statsmodels-0.6.1/debian/python-statsmodels/usr/lib/python2.7/dist-packages/statsmodels/regression/quantile_regression.py:189: ConvergenceWarning: Convergence cycle detected
     warnings.warn(&quot;Convergence cycle detected&quot;, ConvergenceWarning)
   /build/statsmodels-ungkPp/statsmodels-0.6.1/debian/python-statsmodels/usr/lib/python2.7/dist-packages/statsmodels/regression/quantile_regression.py:189: ConvergenceWarning: Convergence cycle detected
     warnings.warn(&quot;Convergence cycle detected&quot;, ConvergenceWarning)
   
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   <h3 id="first-plot">First plot</h3>
   <p>This plot compares best fit lines for 10 quantile regression models to the least squares fit. As Koenker and Hallock (2001) point out, we see that:</p>
   <ol>
   <li>Food expenditure increases with income</li>
   <li>The <em>dispersion</em> of food expenditure increases with income</li>
   <li>The least squares estimates fit low income observations quite poorly (i.e. the OLS line passes over most low income households)</li>
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   <div class="highlight"><pre><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">income</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">data</span><span class="o">.</span><span class="n">income</span><span class="o">.</span><span class="n">max</span><span class="p">(),</span> <span class="mi">50</span><span class="p">)</span>
   <span class="n">get_y</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">*</span> <span class="n">x</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="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">models</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
       <span class="n">y</span> <span class="o">=</span> <span class="n">get_y</span><span class="p">(</span><span class="n">models</span><span class="o">.</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">models</span><span class="o">.</span><span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
       <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;dotted&#39;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;grey&#39;</span><span class="p">)</span>
       
   <span class="n">y</span> <span class="o">=</span> <span class="n">get_y</span><span class="p">(</span><span class="n">ols</span><span class="p">[</span><span class="s1">&#39;a&#39;</span><span class="p">],</span> <span class="n">ols</span><span class="p">[</span><span class="s1">&#39;b&#39;</span><span class="p">])</span>
   
   <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;red&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;OLS&#39;</span><span class="p">)</span>
   <span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">income</span><span class="p">,</span> <span class="n">data</span><span class="o">.</span><span class="n">foodexp</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=.</span><span class="mi">2</span><span class="p">)</span>
   <span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">((</span><span class="mi">240</span><span class="p">,</span> <span class="mi">3000</span><span class="p">))</span>
   <span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">((</span><span class="mi">240</span><span class="p">,</span> <span class="mi">2000</span><span class="p">))</span>
   <span class="n">legend</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
   <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;Income&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
   <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Food expenditure&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">);</span>
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   <h3 id="second-plot">Second plot</h3>
   <p>The dotted black lines form 95% point-wise confidence band around 10 quantile regression estimates (solid black line). The red lines represent OLS regression results along with their 95% confindence interval.</p>
   <p>In most cases, the quantile regression point estimates lie outside the OLS confidence interval, which suggests that the effect of income on food expenditure may not be constant across the distribution.</p>
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   <div class="prompt input_prompt">In&nbsp;[5]:</div>
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   <div class="highlight"><pre><span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">rc</span>
   <span class="n">rc</span><span class="p">(</span><span class="s1">&#39;text&#39;</span><span class="p">,</span> <span class="n">usetex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
   <span class="n">n</span> <span class="o">=</span> <span class="n">models</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
   <span class="n">p1</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">models</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">models</span><span class="o">.</span><span class="n">b</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Quantile Reg.&#39;</span><span class="p">)</span>
   <span class="n">p2</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">models</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">models</span><span class="o">.</span><span class="n">ub</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;dotted&#39;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">)</span>
   <span class="n">p3</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">models</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">models</span><span class="o">.</span><span class="n">lb</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;dotted&#39;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">)</span>
   <span class="n">p4</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">models</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="p">[</span><span class="n">ols</span><span class="p">[</span><span class="s1">&#39;b&#39;</span><span class="p">]]</span> <span class="o">*</span> <span class="n">n</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;red&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;OLS&#39;</span><span class="p">)</span>
   <span class="n">p5</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">models</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="p">[</span><span class="n">ols</span><span class="p">[</span><span class="s1">&#39;lb&#39;</span><span class="p">]]</span> <span class="o">*</span> <span class="n">n</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;dotted&#39;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;red&#39;</span><span class="p">)</span>
   <span class="n">p6</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">models</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="p">[</span><span class="n">ols</span><span class="p">[</span><span class="s1">&#39;ub&#39;</span><span class="p">]]</span> <span class="o">*</span> <span class="n">n</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;dotted&#39;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;red&#39;</span><span class="p">)</span>
   <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">r&#39;\beta_\mbox{income}&#39;</span><span class="p">)</span>
   <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;Quantiles of the conditional food expenditure distribution&#39;</span><span class="p">)</span>
   <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
   <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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