Ordinary Least Squares
========================
.. _ols_notebook:
`Link to Notebook GitHub <https://github.com/statsmodels/statsmodels/blob/master/examples/notebooks/ols.ipynb>`_
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<div class=" highlight hl-ipython3"><pre><span class="kn">from</span> <span class="nn">__future__</span> <span class="k">import</span> <span class="n">print_function</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">statsmodels.api</span> <span class="k">as</span> <span class="nn">sm</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">statsmodels.sandbox.regression.predstd</span> <span class="k">import</span> <span class="n">wls_prediction_std</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">9876789</span><span class="p">)</span>
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<h2 id="OLS-estimation">OLS estimation<a class="anchor-link" href="#OLS-estimation">¶</a></h2><p>Artificial data:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">nsample</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">column_stack</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
<span class="n">beta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</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">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">nsample</span><span class="p">)</span>
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<p>Our model needs an intercept so we add a column of 1s:</p>
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<div class=" highlight hl-ipython3"><pre><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="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>
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<p>Fit and summary:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">model</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="n">results</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">()</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>
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<p>Quantities of interest can be extracted directly from the fitted model. Type <code>dir(results)</code> for a full list. Here are some examples:</p>
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<div class=" highlight hl-ipython3"><pre><span class="nb">print</span><span class="p">(</span><span class="s">'Parameters: '</span><span class="p">,</span> <span class="n">results</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s">'R2: '</span><span class="p">,</span> <span class="n">results</span><span class="o">.</span><span class="n">rsquared</span><span class="p">)</span>
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<pre> OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 1.000
Model: OLS Adj. R-squared: 1.000
Method: Least Squares F-statistic: 4.020e+06
Date: Mon, 20 Jul 2015 Prob (F-statistic): 2.83e-239
Time: 17:44:10 Log-Likelihood: -146.51
No. Observations: 100 AIC: 299.0
Df Residuals: 97 BIC: 306.8
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [95.0% Conf. Int.]
------------------------------------------------------------------------------
const 1.3423 0.313 4.292 0.000 0.722 1.963
x1 -0.0402 0.145 -0.278 0.781 -0.327 0.247
x2 10.0103 0.014 715.745 0.000 9.982 10.038
==============================================================================
Omnibus: 2.042 Durbin-Watson: 2.274
Prob(Omnibus): 0.360 Jarque-Bera (JB): 1.875
Skew: 0.234 Prob(JB): 0.392
Kurtosis: 2.519 Cond. No. 144.
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
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<h2 id="OLS-non-linear-curve-but-linear-in-parameters">OLS non-linear curve but linear in parameters<a class="anchor-link" href="#OLS-non-linear-curve-but-linear-in-parameters">¶</a></h2><p>We simulate artificial data with a non-linear relationship between x and y:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">nsample</span> <span class="o">=</span> <span class="mi">50</span>
<span class="n">sig</span> <span class="o">=</span> <span class="mf">0.5</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="n">nsample</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">column_stack</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">5</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">nsample</span><span class="p">)))</span>
<span class="n">beta</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.02</span><span class="p">,</span> <span class="mf">5.</span><span class="p">]</span>
<span class="n">y_true</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="n">y</span> <span class="o">=</span> <span class="n">y_true</span> <span class="o">+</span> <span class="n">sig</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">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">nsample</span><span class="p">)</span>
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<pre>Parameters: [ 1.34233516 -0.04024948 10.01025357]
R2: 0.999987936503
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<p>Fit and summary:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">res</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="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>
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<p>Extract other quantities of interest:</p>
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<div class=" highlight hl-ipython3"><pre><span class="nb">print</span><span class="p">(</span><span class="s">'Parameters: '</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="nb">print</span><span class="p">(</span><span class="s">'Standard errors: '</span><span class="p">,</span> <span class="n">res</span><span class="o">.</span><span class="n">bse</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s">'Predicted values: '</span><span class="p">,</span> <span class="n">res</span><span class="o">.</span><span class="n">predict</span><span class="p">())</span>
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<pre> OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.933
Model: OLS Adj. R-squared: 0.928
Method: Least Squares F-statistic: 211.8
Date: Mon, 20 Jul 2015 Prob (F-statistic): 6.30e-27
Time: 17:44:10 Log-Likelihood: -34.438
No. Observations: 50 AIC: 76.88
Df Residuals: 46 BIC: 84.52
Df Model: 3
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [95.0% Conf. Int.]
------------------------------------------------------------------------------
x1 0.4687 0.026 17.751 0.000 0.416 0.522
x2 0.4836 0.104 4.659 0.000 0.275 0.693
x3 -0.0174 0.002 -7.507 0.000 -0.022 -0.013
const 5.2058 0.171 30.405 0.000 4.861 5.550
==============================================================================
Omnibus: 0.655 Durbin-Watson: 2.896
Prob(Omnibus): 0.721 Jarque-Bera (JB): 0.360
Skew: 0.207 Prob(JB): 0.835
Kurtosis: 3.026 Cond. No. 221.
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
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<p>Draw a plot to compare the true relationship to OLS predictions. Confidence intervals around the predictions are built using the <code>wls_prediction_std</code> command.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">prstd</span><span class="p">,</span> <span class="n">iv_l</span><span class="p">,</span> <span class="n">iv_u</span> <span class="o">=</span> <span class="n">wls_prediction_std</span><span class="p">(</span><span class="n">res</span><span class="p">)</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">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="s">'o'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">"data"</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_true</span><span class="p">,</span> <span class="s">'b-'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">"True"</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">res</span><span class="o">.</span><span class="n">fittedvalues</span><span class="p">,</span> <span class="s">'r--.'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">"OLS"</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">iv_u</span><span class="p">,</span> <span class="s">'r--'</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">iv_l</span><span class="p">,</span> <span class="s">'r--'</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s">'best'</span><span class="p">);</span>
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<pre>Parameters: [ 0.46872448 0.48360119 -0.01740479 5.20584496]
Standard errors: [ 0.02640602 0.10380518 0.00231847 0.17121765]
Predicted values: [ 4.77072516 5.22213464 5.63620761 5.98658823 6.25643234
6.44117491 6.54928009 6.60085051 6.62432454 6.6518039
6.71377946 6.83412169 7.02615877 7.29048685 7.61487206
7.97626054 8.34456611 8.68761335 8.97642389 9.18997755
9.31866582 9.36587056 9.34740836 9.28893189 9.22171529
9.17751587 9.1833565 9.25708583 9.40444579 9.61812821
9.87897556 10.15912843 10.42660281 10.65054491 10.8063004
10.87946503 10.86825119 10.78378163 10.64826203 10.49133265
10.34519853 10.23933827 10.19566084 10.22490593 10.32487947
10.48081414 10.66779556 10.85485568 11.01006072 11.10575781]
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<h2 id="OLS-with-dummy-variables">OLS with dummy variables<a class="anchor-link" href="#OLS-with-dummy-variables">¶</a></h2><p>We generate some artificial data. There are 3 groups which will be modelled using dummy variables. Group 0 is the omitted/benchmark category.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">nsample</span> <span class="o">=</span> <span class="mi">50</span>
<span class="n">groups</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">nsample</span><span class="p">,</span> <span class="nb">int</span><span class="p">)</span>
<span class="n">groups</span><span class="p">[</span><span class="mi">20</span><span class="p">:</span><span class="mi">40</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">groups</span><span class="p">[</span><span class="mi">40</span><span class="p">:]</span> <span class="o">=</span> <span class="mi">2</span>
<span class="c">#dummy = (groups[:,None] == np.unique(groups)).astype(float)</span>
<span class="n">dummy</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">categorical</span><span class="p">(</span><span class="n">groups</span><span class="p">,</span> <span class="n">drop</span><span class="o">=</span><span class="k">True</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="n">nsample</span><span class="p">)</span>
<span class="c"># drop reference category</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">column_stack</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">dummy</span><span class="p">[:,</span><span class="mi">1</span><span class="p">:]))</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="n">prepend</span><span class="o">=</span><span class="k">False</span><span class="p">)</span>
<span class="n">beta</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="p">]</span>
<span class="n">y_true</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="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">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">nsample</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">y_true</span> <span class="o">+</span> <span class="n">e</span>
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<p>Inspect the data:</p>
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<div class=" highlight hl-ipython3"><pre><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="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="nb">print</span><span class="p">(</span><span class="n">groups</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">dummy</span><span class="p">[:</span><span class="mi">5</span><span class="p">,:])</span>
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<p>Fit and summary:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">res2</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="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>
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<pre>[[ 0. 0. 0. 1. ]
[ 0.40816327 0. 0. 1. ]
[ 0.81632653 0. 0. 1. ]
[ 1.2244898 0. 0. 1. ]
[ 1.63265306 0. 0. 1. ]]
[ 9.28223335 10.50481865 11.84389206 10.38508408 12.37941998]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 2 2 2 2 2 2 2 2 2 2]
[[ 1. 0. 0.]
[ 1. 0. 0.]
[ 1. 0. 0.]
[ 1. 0. 0.]
[ 1. 0. 0.]]
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<p>Draw a plot to compare the true relationship to OLS predictions:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">prstd</span><span class="p">,</span> <span class="n">iv_l</span><span class="p">,</span> <span class="n">iv_u</span> <span class="o">=</span> <span class="n">wls_prediction_std</span><span class="p">(</span><span class="n">res2</span><span class="p">)</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">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="s">'o'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">"Data"</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_true</span><span class="p">,</span> <span class="s">'b-'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">"True"</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">res2</span><span class="o">.</span><span class="n">fittedvalues</span><span class="p">,</span> <span class="s">'r--.'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">"Predicted"</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">iv_u</span><span class="p">,</span> <span class="s">'r--'</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">iv_l</span><span class="p">,</span> <span class="s">'r--'</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">loc</span><span class="o">=</span><span class="s">"best"</span><span class="p">)</span>
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<pre> OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.933
Model: OLS Adj. R-squared: 0.928
Method: Least Squares F-statistic: 211.8
Date: Mon, 20 Jul 2015 Prob (F-statistic): 6.30e-27
Time: 17:44:10 Log-Likelihood: -34.438
No. Observations: 50 AIC: 76.88
Df Residuals: 46 BIC: 84.52
Df Model: 3
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [95.0% Conf. Int.]
------------------------------------------------------------------------------
x1 0.4687 0.026 17.751 0.000 0.416 0.522
x2 0.4836 0.104 4.659 0.000 0.275 0.693
x3 -0.0174 0.002 -7.507 0.000 -0.022 -0.013
const 5.2058 0.171 30.405 0.000 4.861 5.550
==============================================================================
Omnibus: 0.655 Durbin-Watson: 2.896
Prob(Omnibus): 0.721 Jarque-Bera (JB): 0.360
Skew: 0.207 Prob(JB): 0.835
Kurtosis: 3.026 Cond. No. 221.
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
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<h2 id="Joint-hypothesis-test">Joint hypothesis test<a class="anchor-link" href="#Joint-hypothesis-test">¶</a></h2><h3 id="F-test">F test<a class="anchor-link" href="#F-test">¶</a></h3><p>We want to test the hypothesis that both coefficients on the dummy variables are equal to zero, that is, $R \times \beta = 0$. An F test leads us to strongly reject the null hypothesis of identical constant in the 3 groups:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">R</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]</span>
<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">R</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="n">res2</span><span class="o">.</span><span class="n">f_test</span><span class="p">(</span><span class="n">R</span><span class="p">))</span>
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<p>You can also use formula-like syntax to test hypotheses</p>
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<div class=" highlight hl-ipython3"><pre><span class="nb">print</span><span class="p">(</span><span class="n">res2</span><span class="o">.</span><span class="n">f_test</span><span class="p">(</span><span class="s">"x2 = x3 = 0"</span><span class="p">))</span>
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<pre>[[0 1 0 0]
[0 0 1 0]]
<F test: F=array([[ 145.49268198]]), p=1.2834419617281833e-20, df_denom=46, df_num=2>
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<h3 id="Small-group-effects">Small group effects<a class="anchor-link" href="#Small-group-effects">¶</a></h3><p>If we generate artificial data with smaller group effects, the T test can no longer reject the Null hypothesis:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">beta</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.0</span><span class="p">,</span> <span class="mi">10</span><span class="p">]</span>
<span class="n">y_true</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="n">y</span> <span class="o">=</span> <span class="n">y_true</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">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">nsample</span><span class="p">)</span>
<span class="n">res3</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>
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<pre><F test: F=array([[ 145.49268198]]), p=1.2834419617282074e-20, df_denom=46, df_num=2>
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<div class=" highlight hl-ipython3"><pre><span class="nb">print</span><span class="p">(</span><span class="n">res3</span><span class="o">.</span><span class="n">f_test</span><span class="p">(</span><span class="n">R</span><span class="p">))</span>
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<div class=" highlight hl-ipython3"><pre><span class="nb">print</span><span class="p">(</span><span class="n">res3</span><span class="o">.</span><span class="n">f_test</span><span class="p">(</span><span class="s">"x2 = x3 = 0"</span><span class="p">))</span>
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<pre><F test: F=array([[ 1.22491119]]), p=0.30318644106311987, df_denom=46, df_num=2>
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<h3 id="Multicollinearity">Multicollinearity<a class="anchor-link" href="#Multicollinearity">¶</a></h3><p>The Longley dataset is well known to have high multicollinearity. That is, the exogenous predictors are highly correlated. This is problematic because it can affect the stability of our coefficient estimates as we make minor changes to model specification.</p>
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<div class=" highlight hl-ipython3"><pre><span class="kn">from</span> <span class="nn">statsmodels.datasets.longley</span> <span class="k">import</span> <span class="n">load_pandas</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">load_pandas</span><span class="p">()</span><span class="o">.</span><span class="n">endog</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">load_pandas</span><span class="p">()</span><span class="o">.</span><span class="n">exog</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>
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<pre><F test: F=array([[ 1.22491119]]), p=0.30318644106311987, df_denom=46, df_num=2>
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<p>Fit and summary:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">ols_model</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="n">ols_results</span> <span class="o">=</span> <span class="n">ols_model</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="n">ols_results</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
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<h4 id="Condition-number">Condition number<a class="anchor-link" href="#Condition-number">¶</a></h4><p>One way to assess multicollinearity is to compute the condition number. Values over 20 are worrisome (see Greene 4.9). The first step is to normalize the independent variables to have unit length:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">norm_x</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">values</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">name</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
<span class="k">if</span> <span class="n">name</span> <span class="o">==</span> <span class="s">"const"</span><span class="p">:</span>
<span class="k">continue</span>
<span class="n">norm_x</span><span class="p">[:,</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">[</span><span class="n">name</span><span class="p">]</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">name</span><span class="p">])</span>
<span class="n">norm_xtx</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">norm_x</span><span class="o">.</span><span class="n">T</span><span class="p">,</span><span class="n">norm_x</span><span class="p">)</span>
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<pre> OLS Regression Results
==============================================================================
Dep. Variable: TOTEMP R-squared: 0.995
Model: OLS Adj. R-squared: 0.992
Method: Least Squares F-statistic: 330.3
Date: Mon, 20 Jul 2015 Prob (F-statistic): 4.98e-10
Time: 17:44:10 Log-Likelihood: -109.62
No. Observations: 16 AIC: 233.2
Df Residuals: 9 BIC: 238.6
Df Model: 6
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [95.0% Conf. Int.]
------------------------------------------------------------------------------
const -3.482e+06 8.9e+05 -3.911 0.004 -5.5e+06 -1.47e+06
GNPDEFL 15.0619 84.915 0.177 0.863 -177.029 207.153
GNP -0.0358 0.033 -1.070 0.313 -0.112 0.040
UNEMP -2.0202 0.488 -4.136 0.003 -3.125 -0.915
ARMED -1.0332 0.214 -4.822 0.001 -1.518 -0.549
POP -0.0511 0.226 -0.226 0.826 -0.563 0.460
YEAR 1829.1515 455.478 4.016 0.003 798.788 2859.515
==============================================================================
Omnibus: 0.749 Durbin-Watson: 2.559
Prob(Omnibus): 0.688 Jarque-Bera (JB): 0.684
Skew: 0.420 Prob(JB): 0.710
Kurtosis: 2.434 Cond. No. 4.86e+09
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 4.86e+09. This might indicate that there are
strong multicollinearity or other numerical problems.
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<pre>/Users/tom.augspurger/Envs/py3/lib/python3.4/site-packages/scipy/stats/stats.py:1233: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=16
int(n))
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<p>Then, we take the square root of the ratio of the biggest to the smallest eigen values.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">eigs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eigvals</span><span class="p">(</span><span class="n">norm_xtx</span><span class="p">)</span>
<span class="n">condition_number</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">eigs</span><span class="o">.</span><span class="n">max</span><span class="p">()</span> <span class="o">/</span> <span class="n">eigs</span><span class="o">.</span><span class="n">min</span><span class="p">())</span>
<span class="nb">print</span><span class="p">(</span><span class="n">condition_number</span><span class="p">)</span>
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<h4 id="Dropping-an-observation">Dropping an observation<a class="anchor-link" href="#Dropping-an-observation">¶</a></h4><p>Greene also points out that dropping a single observation can have a dramatic effect on the coefficient estimates:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">ols_results2</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="o">.</span><span class="n">ix</span><span class="p">[:</span><span class="mi">14</span><span class="p">],</span> <span class="n">X</span><span class="o">.</span><span class="n">ix</span><span class="p">[:</span><span class="mi">14</span><span class="p">])</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s">"Percentage change %4.2f%%</span><span class="se">\n</span><span class="s">"</span><span class="o">*</span><span class="mi">7</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">([</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">(</span><span class="n">ols_results2</span><span class="o">.</span><span class="n">params</span> <span class="o">-</span> <span class="n">ols_results</span><span class="o">.</span><span class="n">params</span><span class="p">)</span><span class="o">/</span><span class="n">ols_results</span><span class="o">.</span><span class="n">params</span><span class="o">*</span><span class="mi">100</span><span class="p">]))</span>
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<pre>56240.8707446
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<p>We can also look at formal statistics for this such as the DFBETAS -- a standardized measure of how much each coefficient changes when that observation is left out.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">infl</span> <span class="o">=</span> <span class="n">ols_results</span><span class="o">.</span><span class="n">get_influence</span><span class="p">()</span>
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<pre>Percentage change -13.35%
Percentage change -236.18%
Percentage change -23.69%
Percentage change -3.36%
Percentage change -7.26%
Percentage change -200.46%
Percentage change -13.34%
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<p>In general we may consider DBETAS in absolute value greater than $2/\sqrt{N}$ to be influential observations</p>
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<div class=" highlight hl-ipython3"><pre><span class="mf">2.</span><span class="o">/</span><span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">**.</span><span class="mi">5</span>
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<div class=" highlight hl-ipython3"><pre><span class="nb">print</span><span class="p">(</span><span class="n">infl</span><span class="o">.</span><span class="n">summary_frame</span><span class="p">()</span><span class="o">.</span><span class="n">filter</span><span class="p">(</span><span class="n">regex</span><span class="o">=</span><span class="s">"dfb"</span><span class="p">))</span>
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