Autoregressive Moving Average (ARMA): Sunspots data
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.. _tsa_arma_0_notebook:

`Link to Notebook GitHub <https://github.com/statsmodels/statsmodels/blob/master/examples/notebooks/tsa_arma_0.ipynb>`_

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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">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
   <span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">stats</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">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
   
   <span class="kn">import</span> <span class="nn">statsmodels.api</span> <span class="kn">as</span> <span class="nn">sm</span>
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   <div class="highlight"><pre><span class="kn">from</span> <span class="nn">statsmodels.graphics.api</span> <span class="kn">import</span> <span class="n">qqplot</span>
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   <h2 id="Sunpots-Data">Sunpots Data<a class="anchor-link" href="#Sunpots-Data">&#182;</a></h2>
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   <div class="highlight"><pre><span class="k">print</span><span class="p">(</span><span class="n">sm</span><span class="o">.</span><span class="n">datasets</span><span class="o">.</span><span class="n">sunspots</span><span class="o">.</span><span class="n">NOTE</span><span class="p">)</span>
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   ::
   
       Number of Observations - 309 (Annual 1700 - 2008)
       Number of Variables - 1
       Variable name definitions::
   
           SUNACTIVITY - Number of sunspots for each year
   
       The data file contains a &apos;YEAR&apos; variable that is not returned by load.
   
   
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   <div class="highlight"><pre><span class="n">dta</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">sunspots</span><span class="o">.</span><span class="n">load_pandas</span><span class="p">()</span><span class="o">.</span><span class="n">data</span>
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   <div class="highlight"><pre><span class="n">dta</span><span class="o">.</span><span class="n">index</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">Index</span><span class="p">(</span><span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">datetools</span><span class="o">.</span><span class="n">dates_from_range</span><span class="p">(</span><span class="s">&#39;1700&#39;</span><span class="p">,</span> <span class="s">&#39;2008&#39;</span><span class="p">))</span>
   <span class="k">del</span> <span class="n">dta</span><span class="p">[</span><span class="s">&quot;YEAR&quot;</span><span class="p">]</span>
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   <div class="highlight"><pre><span class="n">dta</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">));</span>
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   In&nbsp;[7]:
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   <div class="highlight"><pre><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
   <span class="n">ax1</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">211</span><span class="p">)</span>
   <span class="n">fig</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">graphics</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">plot_acf</span><span class="p">(</span><span class="n">dta</span><span class="o">.</span><span class="n">values</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(),</span> <span class="n">lags</span><span class="o">=</span><span class="mi">40</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">)</span>
   <span class="n">ax2</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">212</span><span class="p">)</span>
   <span class="n">fig</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">graphics</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">plot_pacf</span><span class="p">(</span><span class="n">dta</span><span class="p">,</span> <span class="n">lags</span><span class="o">=</span><span class="mi">40</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">)</span>
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   "
   >
   </div>
   
   </div>
   
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   </div>
   
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   In&nbsp;[8]:
   </div>
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       <div class="input_area">
   <div class="highlight"><pre><span class="n">arma_mod20</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">ARMA</span><span class="p">(</span><span class="n">dta</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">))</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
   <span class="k">print</span><span class="p">(</span><span class="n">arma_mod20</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
   </pre></div>
   
   </div>
   </div>
   </div>
   
   <div class="output_wrapper">
   <div class="output">
   
   
   <div class="output_area"><div class="prompt"></div>
   <div class="output_subarea output_stream output_stdout output_text">
   <pre>
   const                49.659445
   ar.L1.SUNACTIVITY     1.390656
   ar.L2.SUNACTIVITY    -0.688571
   dtype: float64
   
   </pre>
   </div>
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">
   In&nbsp;[9]:
   </div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">arma_mod30</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">ARMA</span><span class="p">(</span><span class="n">dta</span><span class="p">,</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">))</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
   </pre></div>
   
   </div>
   </div>
   </div>
   
   </div>
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   In&nbsp;[10]:
   </div>
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       <div class="input_area">
   <div class="highlight"><pre><span class="k">print</span><span class="p">(</span><span class="n">arma_mod20</span><span class="o">.</span><span class="n">aic</span><span class="p">,</span> <span class="n">arma_mod20</span><span class="o">.</span><span class="n">bic</span><span class="p">,</span> <span class="n">arma_mod20</span><span class="o">.</span><span class="n">hqic</span><span class="p">)</span>
   </pre></div>
   
   </div>
   </div>
   </div>
   
   <div class="output_wrapper">
   <div class="output">
   
   
   <div class="output_area"><div class="prompt"></div>
   <div class="output_subarea output_stream output_stdout output_text">
   <pre>
   2622.63633806 2637.56970317 2628.60672591
   
   </pre>
   </div>
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">
   In&nbsp;[11]:
   </div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="k">print</span><span class="p">(</span><span class="n">arma_mod30</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
   </pre></div>
   
   </div>
   </div>
   </div>
   
   <div class="output_wrapper">
   <div class="output">
   
   
   <div class="output_area"><div class="prompt"></div>
   <div class="output_subarea output_stream output_stdout output_text">
   <pre>
   const                49.749821
   ar.L1.SUNACTIVITY     1.300810
   ar.L2.SUNACTIVITY    -0.508093
   ar.L3.SUNACTIVITY    -0.129650
   dtype: float64
   
   </pre>
   </div>
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
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   In&nbsp;[12]:
   </div>
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       <div class="input_area">
   <div class="highlight"><pre><span class="k">print</span><span class="p">(</span><span class="n">arma_mod30</span><span class="o">.</span><span class="n">aic</span><span class="p">,</span> <span class="n">arma_mod30</span><span class="o">.</span><span class="n">bic</span><span class="p">,</span> <span class="n">arma_mod30</span><span class="o">.</span><span class="n">hqic</span><span class="p">)</span>
   </pre></div>
   
   </div>
   </div>
   </div>
   
   <div class="output_wrapper">
   <div class="output">
   
   
   <div class="output_area"><div class="prompt"></div>
   <div class="output_subarea output_stream output_stdout output_text">
   <pre>
   2619.4036287 2638.07033508 2626.86661351
   
   </pre>
   </div>
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing text_cell rendered">
   <div class="prompt input_prompt">
   </div>
   <div class="inner_cell">
   <div class="text_cell_render border-box-sizing rendered_html">
   <ul>
   <li>Does our model obey the theory?</li>
   </ul>
   </div>
   </div>
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">
   In&nbsp;[13]:
   </div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">durbin_watson</span><span class="p">(</span><span class="n">arma_mod30</span><span class="o">.</span><span class="n">resid</span><span class="o">.</span><span class="n">values</span><span class="p">)</span>
   </pre></div>
   
   </div>
   </div>
   </div>
   
   <div class="output_wrapper">
   <div class="output">
   
   
   <div class="output_area"><div class="prompt output_prompt">
       Out[13]:</div>
   
   
   <div class="output_text output_subarea output_pyout">
   <pre>
   1.9564809773919536
   </pre>
   </div>
   
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing code_cell rendered">
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   <div class="prompt input_prompt">
   In&nbsp;[14]:
   </div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
   <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
   <span class="n">ax</span> <span class="o">=</span> <span class="n">arma_mod30</span><span class="o">.</span><span class="n">resid</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">);</span>
   </pre></div>
   
   </div>
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   "
   >
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   In&nbsp;[15]:
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   <div class="highlight"><pre><span class="n">resid</span> <span class="o">=</span> <span class="n">arma_mod30</span><span class="o">.</span><span class="n">resid</span>
   </pre></div>
   
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   In&nbsp;[16]:
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   <div class="highlight"><pre><span class="n">stats</span><span class="o">.</span><span class="n">normaltest</span><span class="p">(</span><span class="n">resid</span><span class="p">)</span>
   </pre></div>
   
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       Out[16]:</div>
   
   
   <div class="output_text output_subarea output_pyout">
   <pre>
   (49.845020057277331, 1.5006914886180469e-11)
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   In&nbsp;[17]:
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   <div class="highlight"><pre><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
   <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
   <span class="n">fig</span> <span class="o">=</span> <span class="n">qqplot</span><span class="p">(</span><span class="n">resid</span><span class="p">,</span> <span class="n">line</span><span class="o">=</span><span class="s">&#39;q&#39;</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">fit</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
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   In&nbsp;[18]:
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   <div class="highlight"><pre><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
   <span class="n">ax1</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">211</span><span class="p">)</span>
   <span class="n">fig</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">graphics</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">plot_acf</span><span class="p">(</span><span class="n">resid</span><span class="o">.</span><span class="n">values</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(),</span> <span class="n">lags</span><span class="o">=</span><span class="mi">40</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">)</span>
   <span class="n">ax2</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">212</span><span class="p">)</span>
   <span class="n">fig</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">graphics</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">plot_pacf</span><span class="p">(</span><span class="n">resid</span><span class="p">,</span> <span class="n">lags</span><span class="o">=</span><span class="mi">40</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">)</span>
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   "
   >
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   In&nbsp;[19]:
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   <div class="highlight"><pre><span class="n">r</span><span class="p">,</span><span class="n">q</span><span class="p">,</span><span class="n">p</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">acf</span><span class="p">(</span><span class="n">resid</span><span class="o">.</span><span class="n">values</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(),</span> <span class="n">qstat</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
   <span class="n">data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">41</span><span class="p">),</span> <span class="n">r</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">q</span><span class="p">,</span> <span class="n">p</span><span class="p">]</span>
   <span class="n">table</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">data</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s">&#39;lag&#39;</span><span class="p">,</span> <span class="s">&quot;AC&quot;</span><span class="p">,</span> <span class="s">&quot;Q&quot;</span><span class="p">,</span> <span class="s">&quot;Prob(&gt;Q)&quot;</span><span class="p">])</span>
   <span class="k">print</span><span class="p">(</span><span class="n">table</span><span class="o">.</span><span class="n">set_index</span><span class="p">(</span><span class="s">&#39;lag&#39;</span><span class="p">))</span>
   </pre></div>
   
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   <pre>
              AC          Q      Prob(&gt;Q)
   lag                                   
   1    0.009179   0.026286  8.712029e-01
   2    0.041793   0.573039  7.508724e-01
   3   -0.001335   0.573598  9.024489e-01
   4    0.136089   6.408937  1.706193e-01
   5    0.092469   9.111861  1.046847e-01
   6    0.091949  11.793287  6.674241e-02
   7    0.068748  13.297250  6.518874e-02
   8   -0.015020  13.369278  9.975981e-02
   9    0.187592  24.641955  3.393853e-03
   10   0.213718  39.322034  2.229438e-05
   11   0.201082  52.361170  2.344917e-07
   12   0.117182  56.804219  8.574151e-08
   13  -0.014055  56.868355  1.893879e-07
   14   0.015398  56.945595  3.997609e-07
   15  -0.024967  57.149348  7.741383e-07
   16   0.080916  59.296807  6.872063e-07
   17   0.041139  59.853780  1.110926e-06
   18  -0.052021  60.747468  1.548409e-06
   19   0.062496  62.041730  1.831617e-06
   20  -0.010302  62.077018  3.381194e-06
   21   0.074453  63.926689  3.193545e-06
   22   0.124955  69.154802  8.978253e-07
   23   0.093162  72.071062  5.799725e-07
   24  -0.082152  74.346715  4.712968e-07
   25   0.015695  74.430072  8.288956e-07
   26  -0.025037  74.642928  1.367271e-06
   27  -0.125861  80.041164  3.722543e-07
   28   0.053225  81.010000  4.716247e-07
   29  -0.038693  81.523825  6.916588e-07
   30  -0.016904  81.622245  1.151653e-06
   31  -0.019296  81.750958  1.868753e-06
   32   0.104990  85.575078  8.927917e-07
   33   0.040086  86.134577  1.247504e-06
   34   0.008829  86.161820  2.047817e-06
   35   0.014588  86.236458  3.263795e-06
   36  -0.119329  91.248904  1.084451e-06
   37  -0.036665  91.723871  1.521920e-06
   38  -0.046193  92.480517  1.938732e-06
   39  -0.017768  92.592885  2.990676e-06
   40  -0.006220  92.606708  4.696979e-06
   
   </pre>
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   <ul>
   <li>This indicates a lack of fit.</li>
   </ul>
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   <ul>
   <li>In-sample dynamic prediction. How good does our model do?</li>
   </ul>
   </div>
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   In&nbsp;[20]:
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   <div class="highlight"><pre><span class="n">predict_sunspots</span> <span class="o">=</span> <span class="n">arma_mod30</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="s">&#39;1990&#39;</span><span class="p">,</span> <span class="s">&#39;2012&#39;</span><span class="p">,</span> <span class="n">dynamic</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">predict_sunspots</span><span class="p">)</span>
   </pre></div>
   
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   <pre>
   1990-12-31    167.047387
   1991-12-31    140.992925
   1992-12-31     94.858983
   1993-12-31     46.860722
   1994-12-31     11.242379
   1995-12-31     -4.721501
   1996-12-31     -1.167097
   1997-12-31     16.185546
   1998-12-31     39.021780
   1999-12-31     59.449803
   2000-12-31     72.170091
   2001-12-31     75.376728
   2002-12-31     70.436382
   2003-12-31     60.731480
   2004-12-31     50.201662
   2005-12-31     42.075874
   2006-12-31     38.114128
   2007-12-31     38.454492
   2008-12-31     41.963679
   2009-12-31     46.869168
   2010-12-31     51.423156
   2011-12-31     54.399622
   2012-12-31     55.321594
   Freq: A-DEC, dtype: float64
   
   </pre>
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   In&nbsp;[21]:
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   <div class="highlight"><pre><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">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
   <span class="n">ax</span> <span class="o">=</span> <span class="n">dta</span><span class="o">.</span><span class="n">ix</span><span class="p">[</span><span class="s">&#39;1950&#39;</span><span class="p">:]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
   <span class="n">fig</span> <span class="o">=</span> <span class="n">arma_mod30</span><span class="o">.</span><span class="n">plot_predict</span><span class="p">(</span><span class="s">&#39;1990&#39;</span><span class="p">,</span> <span class="s">&#39;2012&#39;</span><span class="p">,</span> <span class="n">dynamic</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">plot_insample</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
   </pre></div>
   
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   In&nbsp;[22]:
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   <div class="highlight"><pre><span class="k">def</span> <span class="nf">mean_forecast_err</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">yhat</span><span class="p">):</span>
       <span class="k">return</span> <span class="n">y</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="n">yhat</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
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   In&nbsp;[23]:
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   <div class="highlight"><pre><span class="n">mean_forecast_err</span><span class="p">(</span><span class="n">dta</span><span class="o">.</span><span class="n">SUNACTIVITY</span><span class="p">,</span> <span class="n">predict_sunspots</span><span class="p">)</span>
   </pre></div>
   
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   5.6370819184412388
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   <h3 id="Exercise:-Can-you-obtain-a-better-fit-for-the-Sunspots-model?-(Hint:-sm.tsa.AR-has-a-method-select_order)">Exercise: Can you obtain a better fit for the Sunspots model? (Hint: sm.tsa.AR has a method select_order)<a class="anchor-link" href="#Exercise:-Can-you-obtain-a-better-fit-for-the-Sunspots-model?-(Hint:-sm.tsa.AR-has-a-method-select_order)">&#182;</a></h3>
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   <h3 id="Simulated-ARMA(4,1):-Model-Identification-is-Difficult">Simulated ARMA(4,1): Model Identification is Difficult<a class="anchor-link" href="#Simulated-ARMA(4,1):-Model-Identification-is-Difficult">&#182;</a></h3>
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   <div class="highlight"><pre><span class="kn">from</span> <span class="nn">statsmodels.tsa.arima_process</span> <span class="kn">import</span> <span class="n">arma_generate_sample</span><span class="p">,</span> <span class="n">ArmaProcess</span>
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   <div class="highlight"><pre><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">1234</span><span class="p">)</span>
   <span class="c"># include zero-th lag</span>
   <span class="n">arparams</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="o">.</span><span class="mi">75</span><span class="p">,</span> <span class="o">-.</span><span class="mi">65</span><span class="p">,</span> <span class="o">-.</span><span class="mi">55</span><span class="p">,</span> <span class="o">.</span><span class="mi">9</span><span class="p">])</span>
   <span class="n">maparams</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="o">.</span><span class="mi">65</span><span class="p">])</span>
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   <p>Let&#39;s make sure this model is estimable.</p>
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   <div class="highlight"><pre><span class="n">arma_t</span> <span class="o">=</span> <span class="n">ArmaProcess</span><span class="p">(</span><span class="n">arparams</span><span class="p">,</span> <span class="n">maparams</span><span class="p">)</span>
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   <div class="highlight"><pre><span class="n">arma_t</span><span class="o">.</span><span class="n">isinvertible</span><span class="p">()</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">TypeError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-132-d3a1a0e5898b&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>arma_t<span class="ansiblue">.</span>isinvertible<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">TypeError</span>: &apos;bool&apos; object is not callable</pre>
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   <div class="highlight"><pre><span class="n">arma_t</span><span class="o">.</span><span class="n">isstationary</span><span class="p">()</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">TypeError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-133-55a9b2cc43b1&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>arma_t<span class="ansiblue">.</span>isstationary<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">TypeError</span>: &apos;bool&apos; object is not callable</pre>
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   <div class="highlight"><pre><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
   <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</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">arma_t</span><span class="o">.</span><span class="n">generate_sample</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">50</span><span class="p">));</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
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   <span class="ansigreen">      1</span> fig <span class="ansiblue">=</span> plt<span class="ansiblue">.</span>figure<span class="ansiblue">(</span>figsize<span class="ansiblue">=</span><span class="ansiblue">(</span><span class="ansicyan">12</span><span class="ansiblue">,</span><span class="ansicyan">8</span><span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      2</span> ax <span class="ansiblue">=</span> fig<span class="ansiblue">.</span>add_subplot<span class="ansiblue">(</span><span class="ansicyan">111</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">----&gt; 3</span><span class="ansired"> </span>ax<span class="ansiblue">.</span>plot<span class="ansiblue">(</span>arma_t<span class="ansiblue">.</span>generate_sample<span class="ansiblue">(</span>size<span class="ansiblue">=</span><span class="ansicyan">50</span><span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue">;</span><span class="ansiblue"></span>
   
   <span class="ansired">TypeError</span>: generate_sample() got an unexpected keyword argument &apos;size&apos;</pre>
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   <div class="highlight"><pre><span class="n">arparams</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="o">.</span><span class="mi">35</span><span class="p">,</span> <span class="o">-.</span><span class="mi">15</span><span class="p">,</span> <span class="o">.</span><span class="mi">55</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">])</span>
   <span class="n">maparams</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="o">.</span><span class="mi">65</span><span class="p">])</span>
   <span class="n">arma_t</span> <span class="o">=</span> <span class="n">ArmaProcess</span><span class="p">(</span><span class="n">arparams</span><span class="p">,</span> <span class="n">maparams</span><span class="p">)</span>
   <span class="n">arma_t</span><span class="o">.</span><span class="n">isstationary</span><span class="p">()</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">TypeError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-135-317f1b2ac56b&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">      2</span> maparams <span class="ansiblue">=</span> np<span class="ansiblue">.</span>array<span class="ansiblue">(</span><span class="ansiblue">[</span><span class="ansicyan">1</span><span class="ansiblue">,</span> <span class="ansicyan">.65</span><span class="ansiblue">]</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      3</span> arma_t <span class="ansiblue">=</span> ArmaProcess<span class="ansiblue">(</span>arparams<span class="ansiblue">,</span> maparams<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">----&gt; 4</span><span class="ansired"> </span>arma_t<span class="ansiblue">.</span>isstationary<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">TypeError</span>: &apos;bool&apos; object is not callable</pre>
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   <div class="highlight"><pre><span class="n">arma_rvs</span> <span class="o">=</span> <span class="n">arma_t</span><span class="o">.</span><span class="n">generate_sample</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span> <span class="n">burnin</span><span class="o">=</span><span class="mi">250</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mf">2.5</span><span class="p">)</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">TypeError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-136-e0a3cc13cb6e&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>arma_rvs <span class="ansiblue">=</span> arma_t<span class="ansiblue">.</span>generate_sample<span class="ansiblue">(</span>size<span class="ansiblue">=</span><span class="ansicyan">500</span><span class="ansiblue">,</span> burnin<span class="ansiblue">=</span><span class="ansicyan">250</span><span class="ansiblue">,</span> scale<span class="ansiblue">=</span><span class="ansicyan">2.5</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">TypeError</span>: generate_sample() got an unexpected keyword argument &apos;size&apos;</pre>
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   <div class="highlight"><pre><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
   <span class="n">ax1</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">211</span><span class="p">)</span>
   <span class="n">fig</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">graphics</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">plot_acf</span><span class="p">(</span><span class="n">arma_rvs</span><span class="p">,</span> <span class="n">lags</span><span class="o">=</span><span class="mi">40</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">)</span>
   <span class="n">ax2</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">212</span><span class="p">)</span>
   <span class="n">fig</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">graphics</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">plot_pacf</span><span class="p">(</span><span class="n">arma_rvs</span><span class="p">,</span> <span class="n">lags</span><span class="o">=</span><span class="mi">40</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">)</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-137-8e761b44cfae&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">      1</span> fig <span class="ansiblue">=</span> plt<span class="ansiblue">.</span>figure<span class="ansiblue">(</span>figsize<span class="ansiblue">=</span><span class="ansiblue">(</span><span class="ansicyan">12</span><span class="ansiblue">,</span><span class="ansicyan">8</span><span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      2</span> ax1 <span class="ansiblue">=</span> fig<span class="ansiblue">.</span>add_subplot<span class="ansiblue">(</span><span class="ansicyan">211</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">----&gt; 3</span><span class="ansired"> </span>fig <span class="ansiblue">=</span> sm<span class="ansiblue">.</span>graphics<span class="ansiblue">.</span>tsa<span class="ansiblue">.</span>plot_acf<span class="ansiblue">(</span>arma_rvs<span class="ansiblue">,</span> lags<span class="ansiblue">=</span><span class="ansicyan">40</span><span class="ansiblue">,</span> ax<span class="ansiblue">=</span>ax1<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      4</span> ax2 <span class="ansiblue">=</span> fig<span class="ansiblue">.</span>add_subplot<span class="ansiblue">(</span><span class="ansicyan">212</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      5</span> fig <span class="ansiblue">=</span> sm<span class="ansiblue">.</span>graphics<span class="ansiblue">.</span>tsa<span class="ansiblue">.</span>plot_pacf<span class="ansiblue">(</span>arma_rvs<span class="ansiblue">,</span> lags<span class="ansiblue">=</span><span class="ansicyan">40</span><span class="ansiblue">,</span> ax<span class="ansiblue">=</span>ax2<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;arma_rvs&apos; is not defined</pre>
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   <ul>
   <li>For mixed ARMA processes the Autocorrelation function is a mixture of exponentials and damped sine waves after (q-p) lags. </li>
   <li>The partial autocorrelation function is a mixture of exponentials and dampened sine waves after (p-q) lags.</li>
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   In&nbsp;[33]:
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   <div class="highlight"><pre><span class="n">arma11</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">ARMA</span><span class="p">(</span><span class="n">arma_rvs</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
   <span class="n">resid</span> <span class="o">=</span> <span class="n">arma11</span><span class="o">.</span><span class="n">resid</span>
   <span class="n">r</span><span class="p">,</span><span class="n">q</span><span class="p">,</span><span class="n">p</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">acf</span><span class="p">(</span><span class="n">resid</span><span class="p">,</span> <span class="n">qstat</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
   <span class="n">data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">41</span><span class="p">),</span> <span class="n">r</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">q</span><span class="p">,</span> <span class="n">p</span><span class="p">]</span>
   <span class="n">table</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">data</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s">&#39;lag&#39;</span><span class="p">,</span> <span class="s">&quot;AC&quot;</span><span class="p">,</span> <span class="s">&quot;Q&quot;</span><span class="p">,</span> <span class="s">&quot;Prob(&gt;Q)&quot;</span><span class="p">])</span>
   <span class="k">print</span><span class="p">(</span><span class="n">table</span><span class="o">.</span><span class="n">set_index</span><span class="p">(</span><span class="s">&#39;lag&#39;</span><span class="p">))</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-138-03653831c71c&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>arma11 <span class="ansiblue">=</span> sm<span class="ansiblue">.</span>tsa<span class="ansiblue">.</span>ARMA<span class="ansiblue">(</span>arma_rvs<span class="ansiblue">,</span> <span class="ansiblue">(</span><span class="ansicyan">1</span><span class="ansiblue">,</span><span class="ansicyan">1</span><span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue">.</span>fit<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      2</span> resid <span class="ansiblue">=</span> arma11<span class="ansiblue">.</span>resid<span class="ansiblue"></span>
   <span class="ansigreen">      3</span> r<span class="ansiblue">,</span>q<span class="ansiblue">,</span>p <span class="ansiblue">=</span> sm<span class="ansiblue">.</span>tsa<span class="ansiblue">.</span>acf<span class="ansiblue">(</span>resid<span class="ansiblue">,</span> qstat<span class="ansiblue">=</span>True<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      4</span> data <span class="ansiblue">=</span> np<span class="ansiblue">.</span>c_<span class="ansiblue">[</span>range<span class="ansiblue">(</span><span class="ansicyan">1</span><span class="ansiblue">,</span><span class="ansicyan">41</span><span class="ansiblue">)</span><span class="ansiblue">,</span> r<span class="ansiblue">[</span><span class="ansicyan">1</span><span class="ansiblue">:</span><span class="ansiblue">]</span><span class="ansiblue">,</span> q<span class="ansiblue">,</span> p<span class="ansiblue">]</span><span class="ansiblue"></span>
   <span class="ansigreen">      5</span> table <span class="ansiblue">=</span> pd<span class="ansiblue">.</span>DataFrame<span class="ansiblue">(</span>data<span class="ansiblue">,</span> columns<span class="ansiblue">=</span><span class="ansiblue">[</span><span class="ansiblue">&apos;lag&apos;</span><span class="ansiblue">,</span> <span class="ansiblue">&quot;AC&quot;</span><span class="ansiblue">,</span> <span class="ansiblue">&quot;Q&quot;</span><span class="ansiblue">,</span> <span class="ansiblue">&quot;Prob(&gt;Q)&quot;</span><span class="ansiblue">]</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;arma_rvs&apos; is not defined</pre>
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   <div class="highlight"><pre><span class="n">arma41</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">ARMA</span><span class="p">(</span><span class="n">arma_rvs</span><span class="p">,</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
   <span class="n">resid</span> <span class="o">=</span> <span class="n">arma41</span><span class="o">.</span><span class="n">resid</span>
   <span class="n">r</span><span class="p">,</span><span class="n">q</span><span class="p">,</span><span class="n">p</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">acf</span><span class="p">(</span><span class="n">resid</span><span class="p">,</span> <span class="n">qstat</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
   <span class="n">data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">41</span><span class="p">),</span> <span class="n">r</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">q</span><span class="p">,</span> <span class="n">p</span><span class="p">]</span>
   <span class="n">table</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">data</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s">&#39;lag&#39;</span><span class="p">,</span> <span class="s">&quot;AC&quot;</span><span class="p">,</span> <span class="s">&quot;Q&quot;</span><span class="p">,</span> <span class="s">&quot;Prob(&gt;Q)&quot;</span><span class="p">])</span>
   <span class="k">print</span><span class="p">(</span><span class="n">table</span><span class="o">.</span><span class="n">set_index</span><span class="p">(</span><span class="s">&#39;lag&#39;</span><span class="p">))</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-139-30d9c2f35894&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>arma41 <span class="ansiblue">=</span> sm<span class="ansiblue">.</span>tsa<span class="ansiblue">.</span>ARMA<span class="ansiblue">(</span>arma_rvs<span class="ansiblue">,</span> <span class="ansiblue">(</span><span class="ansicyan">4</span><span class="ansiblue">,</span><span class="ansicyan">1</span><span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue">.</span>fit<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      2</span> resid <span class="ansiblue">=</span> arma41<span class="ansiblue">.</span>resid<span class="ansiblue"></span>
   <span class="ansigreen">      3</span> r<span class="ansiblue">,</span>q<span class="ansiblue">,</span>p <span class="ansiblue">=</span> sm<span class="ansiblue">.</span>tsa<span class="ansiblue">.</span>acf<span class="ansiblue">(</span>resid<span class="ansiblue">,</span> qstat<span class="ansiblue">=</span>True<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      4</span> data <span class="ansiblue">=</span> np<span class="ansiblue">.</span>c_<span class="ansiblue">[</span>range<span class="ansiblue">(</span><span class="ansicyan">1</span><span class="ansiblue">,</span><span class="ansicyan">41</span><span class="ansiblue">)</span><span class="ansiblue">,</span> r<span class="ansiblue">[</span><span class="ansicyan">1</span><span class="ansiblue">:</span><span class="ansiblue">]</span><span class="ansiblue">,</span> q<span class="ansiblue">,</span> p<span class="ansiblue">]</span><span class="ansiblue"></span>
   <span class="ansigreen">      5</span> table <span class="ansiblue">=</span> pd<span class="ansiblue">.</span>DataFrame<span class="ansiblue">(</span>data<span class="ansiblue">,</span> columns<span class="ansiblue">=</span><span class="ansiblue">[</span><span class="ansiblue">&apos;lag&apos;</span><span class="ansiblue">,</span> <span class="ansiblue">&quot;AC&quot;</span><span class="ansiblue">,</span> <span class="ansiblue">&quot;Q&quot;</span><span class="ansiblue">,</span> <span class="ansiblue">&quot;Prob(&gt;Q)&quot;</span><span class="ansiblue">]</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;arma_rvs&apos; is not defined</pre>
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   <h3 id="Exercise:-How-good-of-in-sample-prediction-can-you-do-for-another-series,-say,-CPI">Exercise: How good of in-sample prediction can you do for another series, say, CPI<a class="anchor-link" href="#Exercise:-How-good-of-in-sample-prediction-can-you-do-for-another-series,-say,-CPI">&#182;</a></h3>
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   In&nbsp;[35]:
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   <div class="highlight"><pre><span class="n">macrodta</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">macrodata</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">macrodta</span><span class="o">.</span><span class="n">index</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">Index</span><span class="p">(</span><span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">datetools</span><span class="o">.</span><span class="n">dates_from_range</span><span class="p">(</span><span class="s">&#39;1959Q1&#39;</span><span class="p">,</span> <span class="s">&#39;2009Q3&#39;</span><span class="p">))</span>
   <span class="n">cpi</span> <span class="o">=</span> <span class="n">macrodta</span><span class="p">[</span><span class="s">&quot;cpi&quot;</span><span class="p">]</span>
   </pre></div>
   
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   <h4 id="Hint:">Hint:<a class="anchor-link" href="#Hint:">&#182;</a></h4>
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   <div class="highlight"><pre><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
   <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
   <span class="n">ax</span> <span class="o">=</span> <span class="n">cpi</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">);</span>
   <span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">();</span>
   </pre></div>
   
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   QmCC
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   <p>P-value of the unit-root test, resoundly rejects the null of no unit-root.</p>
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   In&nbsp;[37]:
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   <div class="highlight"><pre><span class="k">print</span><span class="p">(</span><span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">adfuller</span><span class="p">(</span><span class="n">cpi</span><span class="p">)[</span><span class="mi">1</span><span class="p">])</span>
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   0.990432818834
   
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