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 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">from</span> <span class="nn">scipy</span> <span class="k">import</span> <span class="n">stats</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</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">import</span> <span class="nn">statsmodels.api</span> <span class="k">as</span> <span class="nn">sm</span>
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<div class=" highlight hl-ipython3"><pre><span class="kn">from</span> <span class="nn">statsmodels.graphics.api</span> <span class="k">import</span> <span class="n">qqplot</span>
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<h2 id="Sunpots-Data">Sunpots Data<a class="anchor-link" href="#Sunpots-Data">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="nb">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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<div class=" highlight hl-ipython3"><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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<pre>::
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 'YEAR' variable that is not returned by load.
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<div class=" highlight hl-ipython3"><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">'1700'</span><span class="p">,</span> <span class="s">'2008'</span><span class="p">))</span>
<span class="k">del</span> <span class="n">dta</span><span class="p">[</span><span class="s">"YEAR"</span><span class="p">]</span>
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<div class=" highlight hl-ipython3"><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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<div class=" highlight hl-ipython3"><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 class=" highlight hl-ipython3"><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="nb">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>
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<div class=" highlight hl-ipython3"><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>
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<pre>const 49.659269
ar.L1.SUNACTIVITY 1.390656
ar.L2.SUNACTIVITY -0.688571
dtype: float64
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<div class=" highlight hl-ipython3"><pre><span class="nb">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>
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<div class=" highlight hl-ipython3"><pre><span class="nb">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>
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<pre>2622.6363380653256 2637.56970317 2628.60672591
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<div class=" highlight hl-ipython3"><pre><span class="nb">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>
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<pre>const 49.749903
ar.L1.SUNACTIVITY 1.300810
ar.L2.SUNACTIVITY -0.508092
ar.L3.SUNACTIVITY -0.129650
dtype: float64
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<li>Does our model obey the theory?</li>
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<div class=" highlight hl-ipython3"><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>
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<pre>2619.403628696675 2638.07033508 2626.8666135
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<div class=" highlight hl-ipython3"><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>
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<div class=" highlight hl-ipython3"><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>
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<div class=" highlight hl-ipython3"><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>
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<div class=" highlight hl-ipython3"><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">'q'</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="k">True</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><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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<div class=" highlight hl-ipython3"><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="k">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">'lag'</span><span class="p">,</span> <span class="s">"AC"</span><span class="p">,</span> <span class="s">"Q"</span><span class="p">,</span> <span class="s">"Prob(>Q)"</span><span class="p">])</span>
<span class="nb">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">'lag'</span><span class="p">))</span>
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<li>This indicates a lack of fit.</li>
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<li>In-sample dynamic prediction. How good does our model do?</li>
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<div class=" highlight hl-ipython3"><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">'1990'</span><span class="p">,</span> <span class="s">'2012'</span><span class="p">,</span> <span class="n">dynamic</span><span class="o">=</span><span class="k">True</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">predict_sunspots</span><span class="p">)</span>
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<pre> AC Q Prob(>Q)
lag
1 0.009179 0.026286 8.712027e-01
2 0.041793 0.573037 7.508733e-01
3 -0.001335 0.573596 9.024494e-01
4 0.136089 6.408916 1.706206e-01
5 0.092468 9.111828 1.046860e-01
6 0.091948 11.793245 6.674342e-02
7 0.068748 13.297202 6.518980e-02
8 -0.015020 13.369231 9.976127e-02
9 0.187592 24.641906 3.393914e-03
10 0.213718 39.321990 2.229478e-05
11 0.201082 52.361129 2.344958e-07
12 0.117182 56.804175 8.574305e-08
13 -0.014055 56.868312 1.893912e-07
14 0.015398 56.945551 3.997679e-07
15 -0.024967 57.149306 7.741510e-07
16 0.080916 59.296760 6.872189e-07
17 0.041138 59.853730 1.110947e-06
18 -0.052021 60.747420 1.548437e-06
19 0.062496 62.041682 1.831649e-06
20 -0.010302 62.076969 3.381254e-06
21 0.074453 63.926642 3.193599e-06
22 0.124955 69.154758 8.978395e-07
23 0.093162 72.071020 5.799812e-07
24 -0.082152 74.346673 4.713038e-07
25 0.015695 74.430029 8.289080e-07
26 -0.025037 74.642887 1.367290e-06
27 -0.125861 80.041129 3.722589e-07
28 0.053225 81.009964 4.716306e-07
29 -0.038693 81.523789 6.916674e-07
30 -0.016904 81.622208 1.151668e-06
31 -0.019296 81.750920 1.868776e-06
32 0.104990 85.575045 8.928012e-07
33 0.040086 86.134547 1.247516e-06
34 0.008829 86.161790 2.047837e-06
35 0.014588 86.236427 3.263827e-06
36 -0.119329 91.248877 1.084461e-06
37 -0.036665 91.723845 1.521932e-06
38 -0.046193 92.480493 1.938747e-06
39 -0.017768 92.592861 2.990698e-06
40 -0.006220 92.606684 4.697013e-06
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<div class=" highlight hl-ipython3"><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">'1950'</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">'1990'</span><span class="p">,</span> <span class="s">'2012'</span><span class="p">,</span> <span class="n">dynamic</span><span class="o">=</span><span class="k">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="k">False</span><span class="p">)</span>
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<pre>1990-12-31 167.047417
1991-12-31 140.993004
1992-12-31 94.859113
1993-12-31 46.860889
1994-12-31 11.242556
1995-12-31 -4.721343
1996-12-31 -1.166978
1997-12-31 16.185618
1998-12-31 39.021814
1999-12-31 59.449818
2000-12-31 72.170108
2001-12-31 75.376767
2002-12-31 70.436452
2003-12-31 60.731580
2004-12-31 50.201783
2005-12-31 42.076000
2006-12-31 38.114247
2007-12-31 38.454593
2008-12-31 41.963761
2009-12-31 46.869234
2010-12-31 51.423214
2011-12-31 54.399679
2012-12-31 55.321659
Freq: A-DEC, dtype: float64
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<div class=" highlight hl-ipython3"><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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<div class=" highlight hl-ipython3"><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>
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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)">¶</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">¶</a></h3>
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<div class=" highlight hl-ipython3"><pre><span class="kn">from</span> <span class="nn">statsmodels.tsa.arima_process</span> <span class="k">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 hl-ipython3"><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's make sure this model is estimable.</p>
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<div class=" highlight hl-ipython3"><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 hl-ipython3"><pre><span class="n">arma_t</span><span class="o">.</span><span class="n">isinvertible</span><span class="p">()</span>
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<div class=" highlight hl-ipython3"><pre><span class="n">arma_t</span><span class="o">.</span><span class="n">isstationary</span><span class="p">()</span>
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<li>What does this mean?</li>
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<div class=" highlight hl-ipython3"><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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<div class=" highlight hl-ipython3"><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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<div class=" highlight hl-ipython3"><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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<div class=" highlight hl-ipython3"><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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<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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<div class=" highlight hl-ipython3"><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="k">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">'lag'</span><span class="p">,</span> <span class="s">"AC"</span><span class="p">,</span> <span class="s">"Q"</span><span class="p">,</span> <span class="s">"Prob(>Q)"</span><span class="p">])</span>
<span class="nb">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">'lag'</span><span class="p">))</span>
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<div class=" highlight hl-ipython3"><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="k">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">'lag'</span><span class="p">,</span> <span class="s">"AC"</span><span class="p">,</span> <span class="s">"Q"</span><span class="p">,</span> <span class="s">"Prob(>Q)"</span><span class="p">])</span>
<span class="nb">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">'lag'</span><span class="p">))</span>
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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">¶</a></h3>
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<div class=" highlight hl-ipython3"><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">'1959Q1'</span><span class="p">,</span> <span class="s">'2009Q3'</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">"cpi"</span><span class="p">]</span>
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<h4 id="Hint:">Hint:<a class="anchor-link" href="#Hint:">¶</a></h4>
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<div class=" highlight hl-ipython3"><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>
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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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<div class=" highlight hl-ipython3"><pre><span class="nb">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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<pre>0.990432818834
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