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<!DOCTYPE html>
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<title>Vector Autoregressions tsa.vector_ar — statsmodels 0.15.1 (+9)</title>
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<section id="vector-autoregressions-tsa-vector-ar">
<span id="var"></span><span id="module-statsmodels.tsa.vector_ar.var_model"></span><h1>Vector Autoregressions <code class="xref py py-mod docutils literal notranslate"><span class="pre">tsa.vector_ar</span></code><a class="headerlink" href="#vector-autoregressions-tsa-vector-ar" title="Link to this heading">#</a></h1>
<p><a class="reference internal" href="#module-statsmodels.tsa.vector_ar" title="statsmodels.tsa.vector_ar: Vector autoregressions and related tools"><code class="xref py py-mod docutils literal notranslate"><span class="pre">statsmodels.tsa.vector_ar</span></code></a> contains methods that are useful
for simultaneously modeling and analyzing multiple time series using
<a class="reference internal" href="#var"><span class="std std-ref">Vector Autoregressions (VAR)</span></a> and
<a class="reference internal" href="#vecm"><span class="std std-ref">Vector Error Correction Models (VECM)</span></a>.</p>
<section id="var-p-processes">
<span id="var-process"></span><h2>VAR(p) processes<a class="headerlink" href="#var-p-processes" title="Link to this heading">#</a></h2>
<p>We are interested in modeling a <span class="math notranslate nohighlight">\(T \times K\)</span> multivariate time series
<span class="math notranslate nohighlight">\(Y\)</span>, where <span class="math notranslate nohighlight">\(T\)</span> denotes the number of observations and <span class="math notranslate nohighlight">\(K\)</span> the
number of variables. One way of estimating relationships between the time series
and their lagged values is the <em>vector autoregression process</em>:</p>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}Y_t = \nu + A_1 Y_{t-1} + \ldots + A_p Y_{t-p} + u_t\\u_t \sim {\sf Normal}(0, \Sigma_u)\end{aligned}\end{align} \]</div>
<p>where <span class="math notranslate nohighlight">\(A_i\)</span> is a <span class="math notranslate nohighlight">\(K \times K\)</span> coefficient matrix.</p>
<p>We follow in large part the methods and notation of <a class="reference external" href="https://www.springer.com/gb/book/9783540401728">Lutkepohl (2005)</a>,
which we will not develop here.</p>
<section id="model-fitting">
<h3>Model fitting<a class="headerlink" href="#model-fitting" title="Link to this heading">#</a></h3>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The classes referenced below are accessible via the
<code class="xref py py-mod docutils literal notranslate"><span class="pre">statsmodels.tsa.api</span></code> module.</p>
</div>
<p>To estimate a VAR model, one must first create the model using an <cite>ndarray</cite> of
homogeneous or structured dtype. When using a structured or record array, the
class will use the passed variable names. Otherwise they can be passed
explicitly:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="go"># some example data</span>
<span class="gp">In [1]: </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">In [2]: </span><span class="kn">import</span><span class="w"> </span><span class="nn">pandas</span>
<span class="gp">In [3]: </span><span class="kn">import</span><span class="w"> </span><span class="nn">statsmodels.api</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">sm</span>
<span class="gp">In [4]: </span><span class="kn">from</span><span class="w"> </span><span class="nn">statsmodels.tsa.api</span><span class="w"> </span><span class="kn">import</span> <span class="n">VAR</span>
<span class="gp">In [5]: </span><span class="n">mdata</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="go"># prepare the dates index</span>
<span class="gp">In [6]: </span><span class="n">dates</span> <span class="o">=</span> <span class="n">mdata</span><span class="p">[[</span><span class="s1">'year'</span><span class="p">,</span> <span class="s1">'quarter'</span><span class="p">]]</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">str</span><span class="p">)</span>
<span class="gp">In [7]: </span><span class="n">quarterly</span> <span class="o">=</span> <span class="n">dates</span><span class="p">[</span><span class="s2">"year"</span><span class="p">]</span> <span class="o">+</span> <span class="s2">"Q"</span> <span class="o">+</span> <span class="n">dates</span><span class="p">[</span><span class="s2">"quarter"</span><span class="p">]</span>
<span class="gp">In [8]: </span><span class="kn">from</span><span class="w"> </span><span class="nn">statsmodels.tsa.base.datetools</span><span class="w"> </span><span class="kn">import</span> <span class="n">dates_from_str</span>
<span class="gp">In [9]: </span><span class="n">quarterly</span> <span class="o">=</span> <span class="n">dates_from_str</span><span class="p">(</span><span class="n">quarterly</span><span class="p">)</span>
<span class="gp">In [10]: </span><span class="n">mdata</span> <span class="o">=</span> <span class="n">mdata</span><span class="p">[[</span><span class="s1">'realgdp'</span><span class="p">,</span><span class="s1">'realcons'</span><span class="p">,</span><span class="s1">'realinv'</span><span class="p">]]</span>
<span class="gp">In [11]: </span><span class="n">mdata</span><span class="o">.</span><span class="n">index</span> <span class="o">=</span> <span class="n">pandas</span><span class="o">.</span><span class="n">DatetimeIndex</span><span class="p">(</span><span class="n">quarterly</span><span class="p">)</span>
<span class="gp">In [12]: </span><span class="n">data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">mdata</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">()</span><span class="o">.</span><span class="n">dropna</span><span class="p">()</span>
<span class="go"># make a VAR model</span>
<span class="gp">In [13]: </span><span class="n">model</span> <span class="o">=</span> <span class="n">VAR</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The <a class="reference internal" href="generated/statsmodels.tsa.vector_ar.var_model.VAR.html#statsmodels.tsa.vector_ar.var_model.VAR" title="statsmodels.tsa.vector_ar.var_model.VAR"><code class="xref py py-class docutils literal notranslate"><span class="pre">VAR</span></code></a> class assumes that the passed time series are
stationary. Non-stationary or trending data can often be transformed to be
stationary by first-differencing or some other method. For direct analysis of
non-stationary time series, a standard stable VAR(p) model is not
appropriate.</p>
</div>
<p>To actually do the estimation, call the <cite>fit</cite> method with the desired lag
order. Or you can have the model select a lag order based on a standard
information criterion (see below):</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [14]: </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="mi">2</span><span class="p">)</span>
<span class="gp">In [15]: </span><span class="n">results</span><span class="o">.</span><span class="n">summary</span><span class="p">()</span>
<span class="gh">Out[15]: </span>
<span class="go"> Summary of Regression Results </span>
<span class="go">==================================</span>
<span class="go">Model: VAR</span>
<span class="go">Method: OLS</span>
<span class="go">Date: Sat, 29, Aug, 2026</span>
<span class="go">Time: 13:49:08</span>
<span class="gt">--------------------------------------------------------------------</span>
<span class="n">No</span><span class="o">.</span> <span class="n">of</span> <span class="n">Equations</span><span class="p">:</span> <span class="mf">3.00000</span> <span class="n">BIC</span><span class="p">:</span> <span class="o">-</span><span class="mf">27.5830</span>
<span class="ne">Nobs</span>: 200.000 HQIC: -27.7892
<span class="n">Log</span> <span class="n">likelihood</span><span class="p">:</span> <span class="mf">1962.57</span> <span class="n">FPE</span><span class="p">:</span> <span class="mf">7.42129e-13</span>
<span class="ne">AIC</span>: -27.9293 Det(Omega_mle): 6.69358e-13
<span class="gt">--------------------------------------------------------------------</span>
<span class="n">Results</span> <span class="k">for</span> <span class="n">equation</span> <span class="n">realgdp</span>
<span class="o">==============================================================================</span>
<span class="n">coefficient</span> <span class="n">std</span><span class="o">.</span> <span class="n">error</span> <span class="n">t</span><span class="o">-</span><span class="n">stat</span> <span class="n">prob</span>
<span class="gt">------------------------------------------------------------------------------</span>
<span class="n">const</span> <span class="mf">0.001527</span> <span class="mf">0.001119</span> <span class="mf">1.365</span> <span class="mf">0.172</span>
<span class="n">L1</span><span class="o">.</span><span class="n">realgdp</span> <span class="o">-</span><span class="mf">0.279435</span> <span class="mf">0.169663</span> <span class="o">-</span><span class="mf">1.647</span> <span class="mf">0.100</span>
<span class="n">L1</span><span class="o">.</span><span class="n">realcons</span> <span class="mf">0.675016</span> <span class="mf">0.131285</span> <span class="mf">5.142</span> <span class="mf">0.000</span>
<span class="n">L1</span><span class="o">.</span><span class="n">realinv</span> <span class="mf">0.033219</span> <span class="mf">0.026194</span> <span class="mf">1.268</span> <span class="mf">0.205</span>
<span class="n">L2</span><span class="o">.</span><span class="n">realgdp</span> <span class="mf">0.008221</span> <span class="mf">0.173522</span> <span class="mf">0.047</span> <span class="mf">0.962</span>
<span class="n">L2</span><span class="o">.</span><span class="n">realcons</span> <span class="mf">0.290458</span> <span class="mf">0.145904</span> <span class="mf">1.991</span> <span class="mf">0.047</span>
<span class="n">L2</span><span class="o">.</span><span class="n">realinv</span> <span class="o">-</span><span class="mf">0.007321</span> <span class="mf">0.025786</span> <span class="o">-</span><span class="mf">0.284</span> <span class="mf">0.776</span>
<span class="o">==============================================================================</span>
<span class="n">Results</span> <span class="k">for</span> <span class="n">equation</span> <span class="n">realcons</span>
<span class="o">==============================================================================</span>
<span class="n">coefficient</span> <span class="n">std</span><span class="o">.</span> <span class="n">error</span> <span class="n">t</span><span class="o">-</span><span class="n">stat</span> <span class="n">prob</span>
<span class="gt">------------------------------------------------------------------------------</span>
<span class="n">const</span> <span class="mf">0.005460</span> <span class="mf">0.000969</span> <span class="mf">5.634</span> <span class="mf">0.000</span>
<span class="n">L1</span><span class="o">.</span><span class="n">realgdp</span> <span class="o">-</span><span class="mf">0.100468</span> <span class="mf">0.146924</span> <span class="o">-</span><span class="mf">0.684</span> <span class="mf">0.494</span>
<span class="n">L1</span><span class="o">.</span><span class="n">realcons</span> <span class="mf">0.268640</span> <span class="mf">0.113690</span> <span class="mf">2.363</span> <span class="mf">0.018</span>
<span class="n">L1</span><span class="o">.</span><span class="n">realinv</span> <span class="mf">0.025739</span> <span class="mf">0.022683</span> <span class="mf">1.135</span> <span class="mf">0.257</span>
<span class="n">L2</span><span class="o">.</span><span class="n">realgdp</span> <span class="o">-</span><span class="mf">0.123174</span> <span class="mf">0.150267</span> <span class="o">-</span><span class="mf">0.820</span> <span class="mf">0.412</span>
<span class="n">L2</span><span class="o">.</span><span class="n">realcons</span> <span class="mf">0.232499</span> <span class="mf">0.126350</span> <span class="mf">1.840</span> <span class="mf">0.066</span>
<span class="n">L2</span><span class="o">.</span><span class="n">realinv</span> <span class="mf">0.023504</span> <span class="mf">0.022330</span> <span class="mf">1.053</span> <span class="mf">0.293</span>
<span class="o">==============================================================================</span>
<span class="n">Results</span> <span class="k">for</span> <span class="n">equation</span> <span class="n">realinv</span>
<span class="o">==============================================================================</span>
<span class="n">coefficient</span> <span class="n">std</span><span class="o">.</span> <span class="n">error</span> <span class="n">t</span><span class="o">-</span><span class="n">stat</span> <span class="n">prob</span>
<span class="gt">------------------------------------------------------------------------------</span>
<span class="n">const</span> <span class="o">-</span><span class="mf">0.023903</span> <span class="mf">0.005863</span> <span class="o">-</span><span class="mf">4.077</span> <span class="mf">0.000</span>
<span class="n">L1</span><span class="o">.</span><span class="n">realgdp</span> <span class="o">-</span><span class="mf">1.970974</span> <span class="mf">0.888892</span> <span class="o">-</span><span class="mf">2.217</span> <span class="mf">0.027</span>
<span class="n">L1</span><span class="o">.</span><span class="n">realcons</span> <span class="mf">4.414162</span> <span class="mf">0.687825</span> <span class="mf">6.418</span> <span class="mf">0.000</span>
<span class="n">L1</span><span class="o">.</span><span class="n">realinv</span> <span class="mf">0.225479</span> <span class="mf">0.137234</span> <span class="mf">1.643</span> <span class="mf">0.100</span>
<span class="n">L2</span><span class="o">.</span><span class="n">realgdp</span> <span class="mf">0.380786</span> <span class="mf">0.909114</span> <span class="mf">0.419</span> <span class="mf">0.675</span>
<span class="n">L2</span><span class="o">.</span><span class="n">realcons</span> <span class="mf">0.800281</span> <span class="mf">0.764416</span> <span class="mf">1.047</span> <span class="mf">0.295</span>
<span class="n">L2</span><span class="o">.</span><span class="n">realinv</span> <span class="o">-</span><span class="mf">0.124079</span> <span class="mf">0.135098</span> <span class="o">-</span><span class="mf">0.918</span> <span class="mf">0.358</span>
<span class="o">==============================================================================</span>
<span class="n">Correlation</span> <span class="n">matrix</span> <span class="n">of</span> <span class="n">residuals</span>
<span class="n">realgdp</span> <span class="n">realcons</span> <span class="n">realinv</span>
<span class="n">realgdp</span> <span class="mf">1.000000</span> <span class="mf">0.603316</span> <span class="mf">0.750722</span>
<span class="n">realcons</span> <span class="mf">0.603316</span> <span class="mf">1.000000</span> <span class="mf">0.131951</span>
<span class="n">realinv</span> <span class="mf">0.750722</span> <span class="mf">0.131951</span> <span class="mf">1.000000</span>
</pre></div>
</div>
<p>Several ways to visualize the data using <cite>matplotlib</cite> are available.</p>
<p>Plotting input time series:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [16]: </span><span class="n">results</span><span class="o">.</span><span class="n">plot</span><span class="p">()</span>
<span class="gh">Out[16]: </span><span class="go"><Figure size 1000x1000 with 3 Axes></span>
</pre></div>
</div>
<img alt="_images/var_plot_input.png" src="_images/var_plot_input.png" />
<p>Plotting time series autocorrelation function:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [17]: </span><span class="n">results</span><span class="o">.</span><span class="n">plot_acorr</span><span class="p">()</span>
<span class="gh">Out[17]: </span><span class="go"><Figure size 1000x1000 with 9 Axes></span>
</pre></div>
</div>
<img alt="_images/var_plot_acorr.png" src="_images/var_plot_acorr.png" />
</section>
<section id="lag-order-selection">
<h3>Lag order selection<a class="headerlink" href="#lag-order-selection" title="Link to this heading">#</a></h3>
<p>Choice of lag order can be a difficult problem. Standard analysis employs
likelihood test or information criteria-based order selection. We have
implemented the latter, accessible through the <a class="reference internal" href="generated/statsmodels.tsa.vector_ar.var_model.VAR.html#statsmodels.tsa.vector_ar.var_model.VAR" title="statsmodels.tsa.vector_ar.var_model.VAR"><code class="xref py py-class docutils literal notranslate"><span class="pre">VAR</span></code></a> class:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [18]: </span><span class="n">model</span><span class="o">.</span><span class="n">select_order</span><span class="p">(</span><span class="mi">15</span><span class="p">)</span>
<span class="gh">Out[18]: </span><span class="go"><statsmodels.tsa.vector_ar.var_model.LagOrderResults at 0x7fdc3251c1a0></span>
</pre></div>
</div>
<p>When calling the <cite>fit</cite> function, one can pass a maximum number of lags and the
order criterion to use for order selection:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [19]: </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="n">maxlags</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span> <span class="n">ic</span><span class="o">=</span><span class="s1">'aic'</span><span class="p">)</span>
</pre></div>
</div>
</section>
<section id="forecasting">
<h3>Forecasting<a class="headerlink" href="#forecasting" title="Link to this heading">#</a></h3>
<p>The linear predictor is the optimal h-step ahead forecast in terms of
mean-squared error:</p>
<div class="math notranslate nohighlight">
\[y_t(h) = \nu + A_1 y_t(h − 1) + \cdots + A_p y_t(h − p)\]</div>
<p>We can use the <cite>forecast</cite> function to produce this forecast. Note that we have
to specify the “initial value” for the forecast:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [20]: </span><span class="n">lag_order</span> <span class="o">=</span> <span class="n">results</span><span class="o">.</span><span class="n">k_ar</span>
<span class="gp">In [21]: </span><span class="n">results</span><span class="o">.</span><span class="n">forecast</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">values</span><span class="p">[</span><span class="o">-</span><span class="n">lag_order</span><span class="p">:],</span> <span class="mi">5</span><span class="p">)</span>
<span class="gh">Out[21]: </span>
<span class="go">array([[ 0.00616044, 0.00500006, 0.00916198],</span>
<span class="go"> [ 0.00427559, 0.00344836, -0.00238478],</span>
<span class="go"> [ 0.00416634, 0.0070728 , -0.01193629],</span>
<span class="go"> [ 0.00557873, 0.00642784, 0.00147152],</span>
<span class="go"> [ 0.00626431, 0.00666715, 0.00379567]])</span>
</pre></div>
</div>
<p>The <cite>forecast_interval</cite> function will produce the above forecast along with
asymptotic standard errors. These can be visualized using the <cite>plot_forecast</cite>
function:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [22]: </span><span class="n">results</span><span class="o">.</span><span class="n">plot_forecast</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="gh">Out[22]: </span><span class="go"><Figure size 1000x1000 with 3 Axes></span>
</pre></div>
</div>
<img alt="_images/var_forecast.png" src="_images/var_forecast.png" />
</section>
<section id="module-statsmodels.tsa.vector_ar">
<span id="class-reference"></span><h3>Class Reference<a class="headerlink" href="#module-statsmodels.tsa.vector_ar" title="Link to this heading">#</a></h3>
<div class="pst-scrollable-table-container"><table class="autosummary longtable table">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.vector_ar.var_model.VAR.html#statsmodels.tsa.vector_ar.var_model.VAR" title="statsmodels.tsa.vector_ar.var_model.VAR"><code class="xref py py-obj docutils literal notranslate"><span class="pre">VAR</span></code></a>(endog[, exog, dates, freq, missing])</p></td>
<td><p>Fit VAR(p) process and do lag order selection</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.vector_ar.var_model.VARProcess.html#statsmodels.tsa.vector_ar.var_model.VARProcess" title="statsmodels.tsa.vector_ar.var_model.VARProcess"><code class="xref py py-obj docutils literal notranslate"><span class="pre">VARProcess</span></code></a>(coefs, coefs_exog, sigma_u[, ...])</p></td>
<td><p>Class represents a known VAR(p) process</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.vector_ar.var_model.VARResults.html#statsmodels.tsa.vector_ar.var_model.VARResults" title="statsmodels.tsa.vector_ar.var_model.VARResults"><code class="xref py py-obj docutils literal notranslate"><span class="pre">VARResults</span></code></a>(endog, endog_lagged, params, ...)</p></td>
<td><p>Estimate VAR(p) process with fixed number of lags</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="post-estimation-analysis">
<h3>Post-estimation Analysis<a class="headerlink" href="#post-estimation-analysis" title="Link to this heading">#</a></h3>
<p>Several process properties and additional results after
estimation are available for vector autoregressive processes.</p>
<div class="pst-scrollable-table-container"><table class="autosummary longtable table">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.vector_ar.var_model.LagOrderResults.html#statsmodels.tsa.vector_ar.var_model.LagOrderResults" title="statsmodels.tsa.vector_ar.var_model.LagOrderResults"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LagOrderResults</span></code></a>(ics, selected_orders[, vecm])</p></td>
<td><p>Results class for choosing a model's lag order</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="normality">
<h3>Normality<a class="headerlink" href="#normality" title="Link to this heading">#</a></h3>
<p>As pointed out in the beginning of this document, the white noise component
<span class="math notranslate nohighlight">\(u_t\)</span> is assumed to be normally distributed. While this assumption
is not required for parameter estimates to be consistent or asymptotically
normal, results are generally more reliable in finite samples when residuals
are Gaussian white noise. To test whether this assumption is consistent with
a data set, <a class="reference internal" href="generated/statsmodels.tsa.vector_ar.var_model.VARResults.html#statsmodels.tsa.vector_ar.var_model.VARResults" title="statsmodels.tsa.vector_ar.var_model.VARResults"><code class="xref py py-class docutils literal notranslate"><span class="pre">VARResults</span></code></a> offers the <cite>test_normality</cite> method.</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [23]: </span><span class="n">results</span><span class="o">.</span><span class="n">test_normality</span><span class="p">()</span>
<span class="gh">Out[23]: </span><span class="go"><statsmodels.tsa.vector_ar.hypothesis_test_results.NormalityTestResults at 0x7fdc328f34d0></span>
</pre></div>
</div>
</section>
<section id="whiteness-of-residuals">
<h3>Whiteness of residuals<a class="headerlink" href="#whiteness-of-residuals" title="Link to this heading">#</a></h3>
<p>To test the whiteness of the estimation residuals (this means absence of
significant residual autocorrelations) one can use the <cite>test_whiteness</cite>
method of <a class="reference internal" href="generated/statsmodels.tsa.vector_ar.var_model.VARResults.html#statsmodels.tsa.vector_ar.var_model.VARResults" title="statsmodels.tsa.vector_ar.var_model.VARResults"><code class="xref py py-class docutils literal notranslate"><span class="pre">VARResults</span></code></a>.</p>
<div class="pst-scrollable-table-container"><table class="autosummary longtable table">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.vector_ar.hypothesis_test_results.HypothesisTestResults.html#statsmodels.tsa.vector_ar.hypothesis_test_results.HypothesisTestResults" title="statsmodels.tsa.vector_ar.hypothesis_test_results.HypothesisTestResults"><code class="xref py py-obj docutils literal notranslate"><span class="pre">HypothesisTestResults</span></code></a>(test_statistic, ...)</p></td>
<td><p>Results class for hypothesis tests</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.vector_ar.hypothesis_test_results.CausalityTestResults.html#statsmodels.tsa.vector_ar.hypothesis_test_results.CausalityTestResults" title="statsmodels.tsa.vector_ar.hypothesis_test_results.CausalityTestResults"><code class="xref py py-obj docutils literal notranslate"><span class="pre">CausalityTestResults</span></code></a>(causing, caused, ...[, ...])</p></td>
<td><p>Results class for Granger-causality and instantaneous causality</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.vector_ar.hypothesis_test_results.NormalityTestResults.html#statsmodels.tsa.vector_ar.hypothesis_test_results.NormalityTestResults" title="statsmodels.tsa.vector_ar.hypothesis_test_results.NormalityTestResults"><code class="xref py py-obj docutils literal notranslate"><span class="pre">NormalityTestResults</span></code></a>(test_statistic, ...)</p></td>
<td><p>Results class for the Jarque-Bera-test for nonnormality</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.vector_ar.hypothesis_test_results.WhitenessTestResults.html#statsmodels.tsa.vector_ar.hypothesis_test_results.WhitenessTestResults" title="statsmodels.tsa.vector_ar.hypothesis_test_results.WhitenessTestResults"><code class="xref py py-obj docutils literal notranslate"><span class="pre">WhitenessTestResults</span></code></a>(test_statistic, ...)</p></td>
<td><p>Results class for the Portmanteau-test for residual autocorrelation</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.vector_ar.hypothesis_test_results.ForecastInterval.html#statsmodels.tsa.vector_ar.hypothesis_test_results.ForecastInterval" title="statsmodels.tsa.vector_ar.hypothesis_test_results.ForecastInterval"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ForecastInterval</span></code></a>(point_forecast, forc_lower, ...)</p></td>
<td><p>Result of the module-level <code class="xref py py-func docutils literal notranslate"><span class="pre">forecast_interval</span></code> and <a class="reference internal" href="generated/statsmodels.tsa.vector_ar.var_model.VARProcess.forecast_interval.html#statsmodels.tsa.vector_ar.var_model.VARProcess.forecast_interval" title="statsmodels.tsa.vector_ar.var_model.VARProcess.forecast_interval"><code class="xref py py-meth docutils literal notranslate"><span class="pre">forecast_interval</span></code></a>.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.vector_ar.hypothesis_test_results.ErrorBand.html#statsmodels.tsa.vector_ar.hypothesis_test_results.ErrorBand" title="statsmodels.tsa.vector_ar.hypothesis_test_results.ErrorBand"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ErrorBand</span></code></a>(lower, upper)</p></td>
<td><p>Impulse-response error band, shared by <a class="reference internal" href="generated/statsmodels.tsa.vector_ar.irf.IRAnalysis.err_band_sz1.html#statsmodels.tsa.vector_ar.irf.IRAnalysis.err_band_sz1" title="statsmodels.tsa.vector_ar.irf.IRAnalysis.err_band_sz1"><code class="xref py py-meth docutils literal notranslate"><span class="pre">err_band_sz1</span></code></a>, <a class="reference internal" href="generated/statsmodels.tsa.vector_ar.irf.IRAnalysis.err_band_sz2.html#statsmodels.tsa.vector_ar.irf.IRAnalysis.err_band_sz2" title="statsmodels.tsa.vector_ar.irf.IRAnalysis.err_band_sz2"><code class="xref py py-meth docutils literal notranslate"><span class="pre">err_band_sz2</span></code></a>, <a class="reference internal" href="generated/statsmodels.tsa.vector_ar.irf.IRAnalysis.err_band_sz3.html#statsmodels.tsa.vector_ar.irf.IRAnalysis.err_band_sz3" title="statsmodels.tsa.vector_ar.irf.IRAnalysis.err_band_sz3"><code class="xref py py-meth docutils literal notranslate"><span class="pre">err_band_sz3</span></code></a>, <a class="reference internal" href="generated/statsmodels.tsa.vector_ar.irf.IRAnalysis.errband_mc.html#statsmodels.tsa.vector_ar.irf.IRAnalysis.errband_mc" title="statsmodels.tsa.vector_ar.irf.IRAnalysis.errband_mc"><code class="xref py py-meth docutils literal notranslate"><span class="pre">errband_mc</span></code></a>, <a class="reference internal" href="generated/statsmodels.tsa.vector_ar.svar_model.SVARResults.sirf_errband_mc.html#statsmodels.tsa.vector_ar.svar_model.SVARResults.sirf_errband_mc" title="statsmodels.tsa.vector_ar.svar_model.SVARResults.sirf_errband_mc"><code class="xref py py-meth docutils literal notranslate"><span class="pre">sirf_errband_mc</span></code></a>, and <a class="reference internal" href="generated/statsmodels.tsa.vector_ar.var_model.VARResults.irf_errband_mc.html#statsmodels.tsa.vector_ar.var_model.VARResults.irf_errband_mc" title="statsmodels.tsa.vector_ar.var_model.VARResults.irf_errband_mc"><code class="xref py py-meth docutils literal notranslate"><span class="pre">irf_errband_mc</span></code></a>.</p></td>
</tr>
</tbody>
</table>
</div>
</section>
</section>
<section id="impulse-response-analysis">
<h2>Impulse Response Analysis<a class="headerlink" href="#impulse-response-analysis" title="Link to this heading">#</a></h2>
<p><em>Impulse responses</em> are of interest in econometric studies: they are the
estimated responses to a unit impulse in one of the variables. They are computed
in practice using the MA(<span class="math notranslate nohighlight">\(\infty\)</span>) representation of the VAR(p) process:</p>
<div class="math notranslate nohighlight">
\[Y_t = \mu + \sum_{i=0}^\infty \Phi_i u_{t-i}\]</div>
<p>We can perform an impulse response analysis by calling the <cite>irf</cite> function on a
<cite>VARResults</cite> object:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [24]: </span><span class="n">irf</span> <span class="o">=</span> <span class="n">results</span><span class="o">.</span><span class="n">irf</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
</pre></div>
</div>
<p>These can be visualized using the <cite>plot</cite> function, in either orthogonalized or
non-orthogonalized form. Asymptotic standard errors are plotted by default at
the 95% significance level, which can be modified by the user.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Orthogonalization is done using the Cholesky decomposition of the estimated
error covariance matrix <span class="math notranslate nohighlight">\(\hat \Sigma_u\)</span> and hence interpretations may
change depending on variable ordering.</p>
</div>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [25]: </span><span class="n">irf</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">orth</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="gh">Out[25]: </span><span class="go"><Figure size 1000x1000 with 9 Axes></span>
</pre></div>
</div>
<img alt="_images/var_irf.png" src="_images/var_irf.png" />
<p>Note the <cite>plot</cite> function is flexible and can plot only variables of interest if
so desired:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [26]: </span><span class="n">irf</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">impulse</span><span class="o">=</span><span class="s1">'realgdp'</span><span class="p">)</span>
<span class="gh">Out[26]: </span><span class="go"><Figure size 1000x1000 with 3 Axes></span>
</pre></div>
</div>
<img alt="_images/var_realgdp.png" src="_images/var_realgdp.png" />
<p>The cumulative effects <span class="math notranslate nohighlight">\(\Psi_n = \sum_{i=0}^n \Phi_i\)</span> can be plotted with
the long run effects as follows:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [27]: </span><span class="n">irf</span><span class="o">.</span><span class="n">plot_cum_effects</span><span class="p">(</span><span class="n">orth</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="gh">Out[27]: </span><span class="go"><Figure size 1000x1000 with 9 Axes></span>
</pre></div>
</div>
<img alt="_images/var_irf_cum.png" src="_images/var_irf_cum.png" />
<div class="pst-scrollable-table-container"><table class="autosummary longtable table">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.vector_ar.irf.IRAnalysis.html#statsmodels.tsa.vector_ar.irf.IRAnalysis" title="statsmodels.tsa.vector_ar.irf.IRAnalysis"><code class="xref py py-obj docutils literal notranslate"><span class="pre">IRAnalysis</span></code></a>(model[, P, periods, order, svar, ...])</p></td>
<td><p>Impulse response analysis class.</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="forecast-error-variance-decomposition-fevd">
<h2>Forecast Error Variance Decomposition (FEVD)<a class="headerlink" href="#forecast-error-variance-decomposition-fevd" title="Link to this heading">#</a></h2>
<p>Forecast errors of component j on k in an i-step ahead forecast can be
decomposed using the orthogonalized impulse responses <span class="math notranslate nohighlight">\(\Theta_i\)</span>:</p>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}\omega_{jk, i} = \sum_{i=0}^{h-1} (e_j^\prime \Theta_i e_k)^2 / \mathrm{MSE}_j(h)\\\mathrm{MSE}_j(h) = \sum_{i=0}^{h-1} e_j^\prime \Phi_i \Sigma_u \Phi_i^\prime e_j\end{aligned}\end{align} \]</div>
<p>These are computed via the <cite>fevd</cite> function up through a total number of steps ahead:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [28]: </span><span class="n">fevd</span> <span class="o">=</span> <span class="n">results</span><span class="o">.</span><span class="n">fevd</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">In [29]: </span><span class="n">fevd</span><span class="o">.</span><span class="n">summary</span><span class="p">()</span>
<span class="go">FEVD for realgdp</span>
<span class="go"> realgdp realcons realinv</span>
<span class="go">0 1.000000 0.000000 0.000000</span>
<span class="go">1 0.864889 0.129253 0.005858</span>
<span class="go">2 0.816725 0.177898 0.005378</span>
<span class="go">3 0.793647 0.197590 0.008763</span>
<span class="go">4 0.777279 0.208127 0.014594</span>
<span class="go">FEVD for realcons</span>
<span class="go"> realgdp realcons realinv</span>
<span class="go">0 0.359877 0.640123 0.000000</span>
<span class="go">1 0.358767 0.635420 0.005813</span>
<span class="go">2 0.348044 0.645138 0.006817</span>
<span class="go">3 0.319913 0.653609 0.026478</span>
<span class="go">4 0.317407 0.652180 0.030414</span>
<span class="go">FEVD for realinv</span>
<span class="go"> realgdp realcons realinv</span>
<span class="go">0 0.577021 0.152783 0.270196</span>
<span class="go">1 0.488158 0.293622 0.218220</span>
<span class="go">2 0.478727 0.314398 0.206874</span>